This question strikes very close to the heart of the Protestant Reformation, along with the subsequent generations of the Reformers. This topic mainly revolves around what the Catholic Church defines as the communion of Saints, which is one of the main Catholic theological DNA strands rejected by Protestants. Perhaps the most feasibly specific answer can be narrowed down to the positions held by the Reformers of the Classical Reformation.
Its important to note that there are quite a few categories of Catholic miracles such as Eucharistic, stigmatic, healing, etc.
However, for the sake of conserving time and space, let's focus specifically on the two aspects indicated in the body of your question...
Miraculous intercession of the Saints
What the Catholic Church teaches
A canonized saint, according to Rome, is a Christian who during his/her life lived a life of exemplary holiness. At first martyrs were singled out as such. From the 4th century on, confessors, those who had refused to deny Christ in the face of death, were also recognized. Finally many others who had lived remarkable lives of holiness were admitted to the cultus.
In full consciousness of this communion of the whole Mystical Body of
Jesus Christ, the Church in its pilgrim members, from the very
earliest days of the Christian religion, has honoured with great
respect the memory of the dead; and, "because it is a holy and a
wholesome thought to pray for the dead that they may be loosed from
their sins" (2 Mac. 12:46) she offers her suffrages for them. The
Church has always believed that the apostles and Christ's martyrs, who
gave the supreme witness of faith and charity by the shedding of their
blood, are closely united with us in Christ; she has always venerated
them, together with the Blessed Virgin Mary and the holy angels, with
a special love, and has asked piously for the help of their
intercession. Soon there were added to these others who had chosen to
imitate more closely the virginity and poverty of Christ, and still
others whom the outstanding practice of the Christian virtues and the
wonderful graces of God recommended to the pious devotion and
imitation of the faithful. (Lumen Gentium 7.50)
By the 12th Century popular devotion to saints swelled so incredibly that it became difficult to separate fact from fiction. It was Alexander III, who in 1170 first declared that no one should be venerated as a saint apart from the Roman Church. This found its way into canon law. The process of beatification and canonization was an attempt to regulate the cult of the saints and to remove the abuses.
Beatification is the act by which the Pope permits the restricted public veneration after death of a Christian who lived an exemplary life of holiness. Normally, the person in question had to have performed one attested miracle. Those beatified receive the title "Blessed." Only those beatified are eligible for Sainthood.
Canonization, as mentioned above, is the definitive declaration by the Pope, that a Christian previously beatified has entered into eternal glory and therefore a public cult is established for the new Saint throughout the whole Church. Prior to this declaration, however, is a long legal process which begins at the diocesan level, proceeds to the Congregation for the Causes of the Saints in Rome, and eventually comes before the cardinals and bishops who advise the Pope. Normally one miracle since beatification is required for canonization. There must be a miracle associated with the person's grave or relics, which is said to occur through the intercession of the saint in heaven.
What the Reformers taught
The Protestant Reformation of the sixteenth century, however, involved a radical reaction against the cult of saints and their relics. The Reformers rejected the idea of saintly intercession, which was thought in Lutheran terminology to constitute reliance on works rather than on faith.
Traditional Lutheran theology teaches that saints pray for the Church in general, but are not mediators of miraculous intercession. Martin Luther, the founder of Lutheranism, approved honoring the saints, but condemned with strict sobriety the acceptance and promulgation of miraculous intercession by the Saints in Heaven.
From his commentary on 1st Peter:
Thus Scripture calls us holy while we are still living here on earth,
if we believe. The papists have taken this name away from us and say:
`We should not be holy; only the saints in heaven are holy.' Therefore
we must get the noble name back. You must be holy. But you must be
prepared not to think that you are holy of yourself or on the strength
of your merit. No, you must be holy because you have the Word of God,
because heaven is yours, and because you have become truly pious and
holy through Christ. This you must avow if you want to be a Christian
(Luther's Works 30:7)
Luther is commenting here on the phrase in 1 Peter 1:2, "by the sanctifying work of the Spirit." But his emphasis, as in the previous quote, is on the Scriptural fact that all believers in Jesus Christ, all Christians, are holy, that is, are saints.
In Luther's 1531 Galatian commentary he reflects on the view of saints he previously held.
When I was a monk, I often had a heartfelt wish to see the life and
conduct of at least one saintly man. But meanwhile I was imagining the
sort of saint who lived in the desert and abstained from food and
drink, subsisting on nothing but roots and cold water. I had derived
this notion about unnatural saints from the books not only of the
sophists but even of the fathers . . . But now that the light of truth
is shining, we see with utter clarity that Christ and the apostles
designate as saints, not those who lead a celibate life, are
abstemious, or who perform other works that give the appearance of
brilliance or grandeur, but those who, being called by the Gospel and
baptized, believe that they have been sanctified and cleansed by the
blood of Christ. Thus whenever Paul writes to Christians, he calls
them saints, sons and heirs of God, etc. Therefore saints are all
those who believe in Christ, whether men or women, slaves or free
(Luther's Works 27:81-82)
A little later in the same work...
When we have repudiated this foolish and wicked notion about the name
"saints" which we suppose applies only to the saints in heaven, and on
earth to hermits and monks who perform some sort of spectacular work
let us now learn from the writings of the apostles that all believers
in Christ are saints (LW 27:83)
John Calvin composed a systematic critique of the cult of miraculous relics in the vernacular (Traité des reliques,1547), in which he rejected the veneration of relics on theological grounds.
The Thirty-Nine Articles formulated by the Anglican Church condemned the "invocation of saints" as...
...a fond thing, vainly invented, and grounded upon no warranty of
Scripture, but rather repugnant to the Word of God. (Article XXII)
Theological opposition often turned to violent iconoclasm on the part of Huguenots during the Wars of Religion. Collections of Catholic relics that had been revered as miraculous were destroyed and the statues of the saints in many French churches still bear the scars of attack.
Many Protestant churches strongly reject all saintly intercession, on the basis of verses like 1 Timothy 2:1–5, which says that Jesus is the sole mediator between God and man, as well as Deuteronomy 18:10-11, which seems to forbid invoking departed souls. They also point to the fact that there are no examples in the Bible of anyone requesting the intercession of the dead - Jesus Christ being the lone exception, because he is believed to be alive and resurrected, and because he is believed to be both human and divine.
According to the doctrine of the Roman Catholic Church, the era of public revelation ended with the death of the last living Apostle. A Marian apparition, if deemed genuine by Church authority, is treated as private revelation that may emphasize some facet of the received public revelation for a specific purpose, but it can never add anything new to the deposit of faith. The Church may pronounce an apparition as worthy of belief, but belief is never required by divine faith. The Holy See has officially confirmed the apparitions at Guadalupe, Saint-Étienne-le-Laus, Paris (Rue du Bac, Miraculous Medal), La Salette, Lourdes, Fátima, Pontmain, Beauraing, and Banneux.
The only way a protestant (or anyone else) could justify belief in Marian apparitions is to accept completely the Roman Catholic view of Mary. If these apparitions are authentic and are performed under the auspice of almighty God, then we are dealing with the Mary revealed in Roman Catholic theology. These apparitions do nothing but confirm distinctly Catholic beliefs.
In 1858 Saint Bernadette Soubirous was a 14-year-old shepherd girl who
lived near the town of Lourdes in France. One day she reported a
vision of a miraculous Lady who identified Herself as "the Immaculate
Conception" in subsequent visions. In the second vision she was asked
to return again and she had 18 visions overall. According to Saint
Bernadette, the Lady held a string of Rosary beads and led Saint
Bernadette to the discovery of a buried spring, also requesting that
the local priests build a chapel at the site of the visions and lead
holy processions there.
The messages of these approved apparitions are, for the most part, cut and dry. You won't find many (if any at all) protestants willing to accept any of this. If you do find any...then perhaps you might consider quietly and charitably directing him or her to their nearest parish priest.