I think that faith is both a private and public thing. It is private in the sense that it is "extremely personal" between the individual person and God. There is no relationship more "personal" than the relationship between one person and God. God knows us better than anyone, and better than we even know ourselves. He knows our thoughts, our heart, our secrets, everything.

Even Jesus needed time away from the crowds to hook up with the Father in private prayer. (For example, Luke 5:16.)

"And when you pray, do not be like the hypocrites, for they love to pray standing in the synagogues and on the street corners to be seen by men. I tell you the truth, they have received their reward in full. But when you pray, go into your room, close the door and pray to your Father, who is unseen. Then your Father, who sees what is done in secret, will reward you." (Matthew 6:5-6)

We do go to church to worship God, and ought to praise His name vocally for His glory, not ours. But God already knows our hearts, so it can be said that we are also doing these physical actions for other humans to see and be influenced by, to be instruments or vessels of God, to further His work on earth.

This reminds me of the Parable of the Talents (although perhaps I'm twisting its meaning here) in Matthew 25:14-30. The master gives his servants various amounts of money ("talents". One of the servants "went at once and put his money to work and gained five more." But another servant simply buried his amount in the ground so that he'd be sure to have it later.

So, to apply this to public vs. private faith, it would seem that private faith -- where one's religion is invisible and unknown to everybody else -- is like taking your salvation and putting it in a hole in the ground. You took what you were given and stored it away for safe-keeping, but nobody else benefitted from it and you didn't try to increase the amount. Burying one's faith in a hole like that seems very selfish, the opposite of what Jesus preached.

To me, one of the most compelling reasons to publicly proclaim one's faith is because at the Day of Judgement, Jesus will speak up on our behalf if we are faithful Christians. So, it seems to me that faithful Christians ought to speak up on Jesus' behalf before that Day comes! If we expect Him to speak up for us before the Father, then we ought to speak up for Jesus before the world.

"You are the light of the world. A city on a hill cannot be hidden. Neither do people light a lamp and put it under a bowl. Instead they put it on its stand, and it gives light to everyone in the house. In the same way, let your light shine before men, that they may see your good deeds and praise your Father in heaven." (Matthew 5:14-16)