The Catholic church teaches that only the Church (Magisterium) has the right to interpret Scriptures whereas Protestant churches typically believe that anyone has that right.
What is the argument that supports the idea that anyone can and should interpret Scripture for himself or herself?
The Bible itself -- if you'll pardon my interpretation -- seems to support such a practice.
In the Old Testament, the law is read aloud to the people (Deut 31:10-13; Josh 8:34f; 2Kgs 23:2; Neh 8:1ff) and the king is required to copy out the law so that he may know it and do it (Deut 17:18ff). In both cases, it is read "raw" and the people and rulers are to understand it and do it. Likewise, the Psalms are meant to be sung by the people, and Proverbs is in part a training manual in wisdom for princes to be read and understood.
In the New Testament, Jesus preaches directly to the people, the Apostles and missionaries preach to the people, and Paul intends that his letters be read aloud to the people (Col 4:16 -- possibly with the courier providing some additional clarification).
Now these are authoritative teachers teaching, so one could argue they are similar in function to the Magisterium, but the "noble Bereans" are also commended for independently searching the scriptures to confirm Paul's teaching (Acts 17:11); Priscilla and Aquila, who are certainly not in the Magisterium, tutor Apollos (Acts 18:24ff); and Luke-Acts was written for the private individual Theophilus (Luke 1:1ff; Acts 1:1).
None of this is to say there is not a binding authority given to the church; there certainly is -- even in Protestantism (how else could church discipline exist as one of the "true marks" of the church?). But individuals are still encouraged to read, hear, understand, and apply the words of the Bible themselves.
In Richard Pratt's formulation of Reformed hermeneutics, IIRC, there are three sources of interpretation one seeks to respect: historical (the church of the past, tradition), community (the church of the present, your pastor and elders), and individual ("private interpretation", conscience, internal illumination by the Holy Spirit). In "remnant" situations, the individual interpretation may be the most correct, and God expects his people to abide by his true word, regardless of what the corrupt authorities say (cf. the OT prophets, and Jesus and John the Baptist vs. the scribes and Pharisees in the NT). That being said, going against broad agreement in interpretation across history and in one's present community should be done only with great prayer, humility, self re-examination, and trepidation.
A footnote on 2 Pet. 1:20 ("no prophecy of Scripture comes from someone’s own interpretation"), a contested verse in this context: "Prophecy" here is effectively "God's communication." Reading the surrounding verses shows (I think!) that the passage is concerned with the origin of a prophecy, specifically that it does not come from a private or merely human source. It is not addressing interpretation or understanding of that divine communication, which is what we're talking about in this question.
When Jesus Died on the Cross, the veil was torn in the Temple, meaning that we do not need a high priest to intercede for us, to God, anymore. Jesus actually does that interceding for us now, He is our high priest.
this coupled with the Holy Ghost (Holy Spirit) means that we can interpret the scriptures for ourselves. we can interpret the scriptures for our lives, to interpret the scriptures for the lives of others is still in the ballpark of those that spend their lives completely dedicated to studying the scripture and being in Prayer and fasting for days at a time, like, say, the Pope.
so really it is a two sided issue.
you can interpret the scriptures for your life, but when you start doing it for others it becomes a delicate subject of whether or not God is really saying that or if you are laying it on someone instead of praying it on someone.
If we do not personally examine the scriptures, how can we then take them to heart? There are many arguments to be made from the scriptures. I will start with just simple reasoning.
Through daily bible reading we reinforce the information we have regarding all the scriptures. If we know the scriptures well, then we know what we are commanded to do, whats expected of us, and the qualities we are to display as Christians.
Now for a scriptural basis for personal interpretation.
Paul recognized in his second letter to Timothy at 2 Ti 4:3-5 that:
"For there will be a period of time when they will not put up with the wholesome teaching, but according to their own desires, they will surround themselves with teachers to have their ears tickled.4 They will turn away from listening to the truth and give attention to false stories.5 You, though, keep your senses in all things, endure hardship, do the work of an evangelizer, fully accomplish your ministry."
What Paul is saying here is that there would come a time when the truth would be twisted. They would teach things that would tickle their ears, or that is things that would be taught according to their own desires. By allowing people to read the scriptures as they please, they find truths in the scriptures.
Watchtower 2005 states:
"Religious leaders tickle people’s ears by condoning practices that appeal to wrong desires, such as sex outside of marriage, homosexuality, and drunkenness. The Bible clearly states that those who approve of such things and those who practice them “will not inherit God’s kingdom.”—1 Corinthians 6:9, 10; Romans 1:24-32."
Do we not see this today? A prime example is the acceptance of homosexuals in the church. If we did not know, from our own reading of the scriptures, Gods commandments on the matter then we would just go along with it.
Another example is from 1st century Christendom. The article is from the 2013 Watchtower article: Receive Full Benefit From Reading the Bible.
"Apollos was a Jewish Christian who was “well versed in the Scriptures” and “aglow with the spirit.” The book of Acts relates about him: “He went speaking and teaching with correctness the things about Jesus, but being acquainted with only the baptism of John.” Without realizing it, Apollos had been teaching an out-of-date understanding of baptism. After hearing him teach in Ephesus, a Christian couple named Priscilla and Aquila explained “the way of God more correctly to him.” (Acts 18:24-26) How did this benefit Apollos?"
Like Apollos, we today may be well versed in scripture. In the case of the above account Apollos was wrong, but how would he have known it if someone with a better understanding didn't come along and adjust his knowledge?
Studying the bible and having our understanding adjusted not only benefits us. In the next verses of acts we see that with Apollos's new information, he was able to teach the baptism accurately. Acts 18:27-28.
We see then that there are many benefits to daily reading and interpreting the bible on our own.
The church is not a mere building where worship happens. The church in the Bible is the people of God (Matthew 16:18).
The church is the support of truth as well as the solid ground of truth (1 Timothy 3:15). This means that the people of God holds the truth and keeps it stedfastly.
but in case I am delayed, I write so that you will know how one ought to conduct himself in the household of God, which is the church of the living God, the pillar and support of the truth.
1 Timothy 3:15 (NASB)
The God-breathed Scripture is God’s truth and it benefits us to do all kinds of good works (John 17:17; cf. 2 Timothy 3:16-17).
In fact, 2 Peter 1:3 tells us that Jesus gave God’s people everything it needs to live a godly life.That includes authority to use the Bible as the sole source of church standards in both doctrine and discipline (2 Timothy 3:16-17).
All Scripture is [a]inspired by God and profitable for teaching, for reproof, for correction, for [b]training in righteousness; 17 so that the man of God may be adequate, equipped for every good work.
2 Timothy 3:16-17 (NASB)
Private interpretation or personal interpretation is only unbiblical when it contradicts the Bible. Thus, 2 Peter 1:20 does not contradict 1 Timothy 2:15. The work of the Christian is to handle the word of truth accurately. Private interpretation is mishandling of the word of truth.
Therefore, a body of Magisterium is not a requirement for church doctrine and church discipline since it tries to exclude the rest of the believers to handle the word of truth. We have come to know that the very idea of a Magisterial body is in fact a practice of private interpretation itself since only a few interprets the Bible and the rest ought to follow it. The Pope speakng “ex chathedra” is also a form of personal interpretation since a singular person is doing interpretation privately, that is, personally.
SUMMARY
The church is the support of truth and solid ground of truth (1 Timothy 3:15). That emphatically shows us that every Christian holds the truth and is a sure foundation of it.Thus, each Christian could handle the word of truth accurately (2 Timothy 2:15). The word of truth is the God-breathed Scripture that is useful for every good works (2 Timothy 3:16-17). In conclusion, although each one can interpret the Scripture as he sees it, there can be no personal interpretation which is at odds with Scripture (2 Peter 1:20; cf. 2 Tim 3:16-17) since the faith is one (Ephesians 4:5) and the church one as well (Ephesians 4:4).
The interpretation of scripture has always been a prerogative of educated clergy and laymen (able ministers - see below.) If one is uneducated in Theology, then any personal interpretation is fraught with difficulties. Peter warns readers.
“As also in all his epistles, speaking in them of these things; in which are some things hard to be understood, which they that are unlearned and unstable wrest, as they do also the other scriptures, unto their own destruction.” 2 Peter 3:16, KJV.
Paul warns us: “Who also hath made us able ministers of the new testament; not of the letter, but of the spirit: for the letter killeth, but the spirit giveth life.” 2 Corinthians 3:6, KJV.
To be safe, an interpreter should refer back to the writings of the Church Fathers, in order to confirm any personal understanding, and if there is a disagreement, then the personal interpretation is probably in error and not safe to teach others. The opinions of uneducated people (not trained in Hebrew, Greek and Latin) cannot be relied upon to accurately represent the meaning of the text. English Bibles disagree with one another on basic Christology, and interpretations will differ because of this.
In the Catholic churches (Anglican, Greek Orthodox, Eastern Orthodox, Syrian, and Roman) the Church Fathers are the source for interpretation of the text. Their councils since the 4th. cent. have been decisive in determining the Christology of the Church. In the confessions of the various Catholic churches the Church Fathers are considered to be the source for the Doctrine of the Church. Everything that has been told about Christ comes from the writings of the Church Fathers and the official documents of the Church.
From the Archbishop and Metropolitan of the Worldwide Anglican Catholic Church:
We are, therefore, in principle, not a sect but a Church, with a universal mission. In fact, as Anglicans we are less sectarian in fundamental impulse than almost any other Christian body: for we firmly assert that while our mission is universal, the particular forms of our own Anglican worship and our own Anglican culture are not exclusive. We deny that we have any unique Anglican Catholic doctrine, but rather we stand for the unique authority of the patristic witness and the Conciliar tradition, and we assert the incompleteness of the Roman Catholic and Eastern Orthodox Churches in their more exclusivist claims. We are Catholic, but do not claim to be the only Catholics. We are Orthodox, but do not claim that in principle all others are heretics. We have clarity of doctrine, but a clarity that embraces East and West, Rome and Orthodox, past and present. ... Our primary goal is not to preserve a tradition, but to share it, even as we recognize that we will have nothing to share if we do not preserve intact what we have received.
The ACC confession of faith: We declare this church to be, and desire that it shall continue, in full Communion with all Anglicans throughout the world who remain faithful to Apostolic Order (including the male Episcopate, Priesthood, and Diaconate), as an integral portion of the one Body of Christ composed of Churches which, united under the One Divine Head and in fellowship of the One Holy Catholic and Apostolic Church, hold the one Faith revealed in Holy Writ, and defined in the Creeds as maintained by the undivided Primitive Catholic Church in the Seven Ecumenical Councils; receive the same Canonical Scriptures of the Old and New Testaments, as containing all things necessary to salvation; teach the same Word of God; partake of the same Divinely ordained Sacraments through the Ministry of the same Divinely instituted Apostolic Orders; and worship one God and Father through the same Lord Jesus Christ, by the same Holy and Divine Spirit Who is given to them that believe to guide them into all truth.
The Anglican Catholic Church is a constituent member of the One Holy Catholic and Apostolic Church of Jesus Christ, faithfully continuing the Anglican tradition. This Church upholds the historic Catholic Faith, Apostolic Order, Orthodox Worship, and Evangelical Witness as set forth in the 1962 Canadian and 1928 American editions of the Book of Common Prayer, and accepts as binding and unalterable the received Faith and Traditions of the Church, and its teachings, including the male Episcopate, Priesthood, and Diaconate, as set forth in the Holy Scriptures; the Apostles', Nicene, and Athanasian Creeds; the writings of the "ancient Catholic Bishops and Doctors"; and especially as defined by the Seven Ecumenical Councils of the Undivided Church.
SECTION 4. OF TRANSLATIONS OF HOLY SCRIPTURE. The Authorised (King James) Version is the received Standard English translation of the Holy Scripture (Old Testament, New Testament, Apocrypha).
We acknowledge that rule of faith laid down by St. Vincent of Lerins: “Let us hold that which has been believed everywhere, always and by all, for that is truly and properly Catholic”.The received Tradition of the Church and its teaching as set forth by “the ancient catholic bishops and doctors”, and especially as defined by the Seven Ecumenical Councils of the undivided Church, to the exclusion of all errors, ancient and modern.
