Catholics believe that one should confess ones sins to God. In private confession with a priest, you are still confessing to God, and it is still God who absolves you. The priest is there to help you, and to hold you accountable. He may give you a penance, which is a way for you to make amends. For example, if your sin was lying to someone, he will remind you that you must come clean and tell the truth to that person. Most sins are involved with allowing your mind to stray away from God, which is why a priest may tell you to pray, so that your mind can stay focused on God.
I don't know any Christian denomination that doesn't have some form of communal, public confession, where the individuals confess sins, and the pastor, preacher, or minister pronounces a general absolution. The difference is, in private confession, you confess specific sins, whereas in the public confession, it's just general.
I would imagine that any pastor, preacher, or minister would be willing to hear a confession of a member of his or her church. I know it is done in many non-Catholic denominations, but it isn't as common as it is among Catholics.
Does that answer your questions?