Group prayer is powerful. Group prayer is natural. Group prayer is necessary.
Christians are to grow in their faith by being in a community of believers. "Lone Ranger Christians" are out of place both in the body of Christ and in a local body of believers. We need one another. We complement one another. We exercise our spiritual gifts in the context and under the authority of an assembly's leadership, whatever form that might take (e.g., elders, deacons, or pastors/ministers/priests/bishops/rectors, to name but a few). Even--or perhaps especially--those with the gift of evangelist, while their ministry is out in the world, need the prayers and fellowship of other believers if they want their gift to be blessed by God.
Paul's analogy between our physical bodies and the body of Christ (both universal and local) is but one approach in answering your question about communal prayer (see 1 Corinthians 12:12-31, especially vv.12-14). Some believers are particularly gifted in public prayer. They have the gift of pouring out their hearts to God in a way which honors God and blesses and encourages fellow believers.
Some believers are especially gifted to be prayer warriors within their private prayer closets. They are every bit as important as the public pray-ers. Frankly, we need both.
Even if you are not particularly gifted in public prayer, whether in the company of one or one hundred other believers, mere self-consciousness should not keep you from doing so. God is our primary audience for prayer. Paul tells us in Romans,
"In the same way the Spirit also helps our weakness; for we do not know how to pray as we should, but the Spirit Himself intercedes for us with groanings too deep for words; and He who searches the hearts knows what the mind of the Spirit is, because He intercedes for the saints according to the will of God" (vv.26-27 NIV).
Public prayer's benefit to those who listen should not be minimized, in my opinion. Even the most "stumbling, bumbling" prayer (if there is such a thing) can be a blessing of encouragement to our fellow Christians. In a small group of two or three, prayer can not only bless and encourage others, but it can bind Christians together more closely and cause them to be more open and even accountable to one another. These are good things which are not to been gainsaid.
There is a touching account of Paul and his fellow missionaries whose ship docked for a short time in Tyre, Syria. The first thing they did was to "look up" the disciples in that city, after which they stayed with them for a week. The missionaries' stay in Tyre ends with the following verse:
"When our days there were ended, we left and started on our journey, while they all, with wives and children, escorted us until we were out of the city. After kneeling down on the beach and praying, we said farewell to one another" (Acts 21:5).
In fact, if you were to take the time to search the book of Acts for the words pray, prayer, praying, and prayed (at biblestudytools.com or biblegateway.com), I feel confident you would realize how important communal prayer was to the early Christians, with so many wonderful things happening as a result of earnest, concerted prayer. Peter's miraculous release from prison is just one such account.
"So Peter was kept in the prison, but prayer for him was being made fervently by the church to God. On the very night when Herod was about to bring him forward, Peter was sleeping [!] between two soldiers, bound with two chains, and guards in front of the door were watching over the prison. And behold, an angel of the Lord suddenly appeared and a light shone in the cell; and he struck Peter's side and woke him up, saying, 'Get up quickly.' And his chains fell off his hands" (Acts 12:5-7, my emphasis).
We know what happened next. The angel led Peter out of the prison, past two more guards, and opened the iron gate that led to the city ("which opened for them by itself," v.10) so that Peter could once again join his brothers and sisters who had been praying for his release!
This account is but one in the book of Acts of the power and necessity of prayer, particularly when a brother or sister is in "crisis mode."
We also need to pray for one another (and by the way, the "one anothers" of the New Testament are good to look up, and not only when researching communal prayer) in non-crisis mode, as when a brother or sister in Christ has an important decision to make, when they have committed a sin, when they are ill, when they are out of work, or when relatives are unsaved (see, for example, James 5:14, 16). The list goes on and on. Just as we are individually to
"pray without ceasing," (1 Thessalonians 5:17)
so also are we to persevere in prayer with our fellow believers. Prayer is the life-blood of the church universal and the church local. Two or more local churches, even of different denominations, should also join together in prayer whenever there is a concerted evangelistic effort, for example. Regardless of denomination, if we name the name of Christ we are one in Christ and should be able to unite with each other in prayer on special occasions.
Individual, private prayer is good. Communal or group prayer is also good. We need not choose between one or the other; rather, we need to do both on a regular basis. Only then will both our personal relationship with God be strengthened and our relationships with other believers become more meaningful, fulfilling, and loving.
Prayer changes things. It also changes people!