"When you encounter a private tree with DNA Contacts does it give the name of the person in their tree that you connect with or do you just get the tree owners profile?"
No, it does not give the person! Even when there is a green hint flag, it does not tell us anything. I have not had much luck with people with private trees answering so far, even when there is a green hint flag. One suggestion I made on the feedback is that we be able to see THAT green hint person, even if we can't see the rest of the tree. That would at least give a clue whether to pursue the hint further or not.