You question is an excellent one. I think I can help you with scripture for public pray and also answer your question about the emphasis Christ made on praying in secret. Prayer is certainly changed according to who is praying and who is listening to the prayer. Though there seem to be many variations, I find it helpful to divide the kinds of prayer into three: private, public, and participatory or mutual. We will look at each kind of prayer.
The most important and the most earnest kind of prayer is that which is done privately. You mention the emphasis Christ made on secret prayer. In Matthew 6:6, Christ states, "But thou, when thou prayest, enter into thy closet, and when thou hast shut thy door, pray to thy Father which is in secret; and thy Father which seeth in secret shall reward thee openly." This certainly shows us the importance of private prayer.
However, Christ is also dealing with a particular practice of the Pharisees. They made a big show of praying so that everyone would know just how spiritual they were. They were proud and ostentatious in their prayers so that others would see them. That is, they took what should have been their private prayer life and made a public show of it so that others would be amazed at their spirituality. This is akin today to those who continually brag on how much time they spend in prayer. Christ was teaching us that our personal prayer life is not to be displayed in public. He was not teaching that there was never an occasion for public prayer.
Public prayer is common in the Old Testament. Solomon prayed at the dedication of the temple (1Kings 8:22-23). Elijah prayed publicly on Mt. Carmel (1Kings 17:36-37). Ezra prayed before "a very great congregation of men and women and children" (Ezra 10:1). If public prayer is not allowed in the New Testament, it is definitely a change in what God allows.
However, we continue to see public prayer practiced in the New Testament even after the teaching of Matthew 6:6. Christ prayed publicly before He raised Lazarus from the dead (John 11:41-42) as well as on other occasions. Paul kneeled and prayed with the Ephesian elders before he left them (Acts 20:36). And although we receive few clear statements about the order of worship in the early churches, an important piece of information is found in 1Corinthians 14:15-16, which states:
1Corinthians 14:15-16 What is it then? I will pray with the spirit, and I will pray with the understanding also: I will sing with the spirit, and I will sing with the understanding also. Else when thou shalt bless with the spirit, how shall he that occupieth the room of the unlearned say Amen at thy giving of thanks, seeing he understandeth not what thou sayest?
If you carefully read this passage, you will see that Paul stresses the importance of praying with understanding so that those who occupy the room of the unlearned can say Amen at the giving of thanks. This passage makes sense only in the context of public prayer. If no one is listening, then how could anyone say Amen? Other statements hint at the presence of public prayer in the early churches, but this one clearly shows that it was practiced.
One thing that might help you is to understand that the purposes of public prayer are a bit different from those of private prayer. In private prayer, we pour our heart out to God and tell Him all. We do not need to be concerned about sentence structure or form in any way. Just talk to God. However, in public prayer, we are leading (as we speak of someone "leading in prayer"); we are leading others to look on God. We are directing the attention of others to the Lord and helping them rely on Him at this time. Again, the sin of Matthew 6:6 was to make a public display of private prayer. It is not dealing with the proper use of public prayer (though public prayer can be misused as a display too).
This could also be called mutual prayer. It refers to times when more than one person participate in prayer together. In one sense, it is public. But instead of one person praying and others saying Amen to their prayers, it involves a group of people joining together in prayer. Yet, there seem to be varying degrees of this. On one end of the spectrum, Jesus took three of His disciples with Him to pray in the Garden of Gethsemane. After the disciples fell asleep, Christ scolded Peter, "What, could ye not watch with me one hour?" (Matthew 26:40). Jesus was going to pray a little distance from them, but they were to watch with Him in prayer. In a sense, they were to participate in His private prayer. On the other end, on one occasion the apostles lift up their voice to God with one accord (Acts 4:24-30). It is almost as if they prayed the same thing in unison.
However, there must have been many variations of this. In Acts 12:5, Peter was thrown into prison, "but prayer was made without ceasing of the church unto God for him." When he is released, he goes to the house of Mary the mother of John Mark, "where many were gathered together praying" (Acts 12:12). This is not private prayer, but neither is it many people listening to one person praying and saying Amen. I believe it was one of the keys of the power of the early churches.
There are other evidences of this participatory prayer in the New Testament. The early disciples met in the upper room and "continued with one accord in prayer and supplication" (Acts 1:14). Paul pleaded with the Romans, "that ye strive together with me in your prayers to God for me" (Romans 15:30). The Corinthians are expected to be "helping together by prayer for us" (2Corinthians 1:11). Taken together, it is certain that the early churches gathered together for mutual prayer in which all participated. We should be doing the same today.
In conclusion, prayer takes many forms. Our truest prayer is to be found in our private times with God. This prayer should be done in the closet and in secret. It is not for display. However, there are proper times for one person leading in public prayer and there are also proper times for God's people to prayer together. Each kind of prayer can find its God-given place in our service to Him.