Just because you've heard gossip, doesn't necessarily mean you have to pass it along to its target
Dear Dr. Levine,
I have a friend who thinks she's being a friend by telling me what other people say about me. I am not talking about compliments - I'm talking about rumors and gossip. I have told her that I don't want to hear it and that it only hurts my feelings. Nonetheless, she insists and says that she'd want to know what people were saying about her. Am I being too sensitive?
You aren't being too sensitive. Since you've told your friend that you don't want to hear negative gossip about yourself, it's somewhat hostile for her to continue to pass on these negative comments. The fact that she would want to know has no bearing on what you want to know.
Another consideration: If she is conveying other people's gossip to you, she may be gossiping to others about things you have said to her. On both accounts, I would be cautious about this friendship.
The only caveat might be if you think the type of information or rumors she is conveying to you could be helpful for you to know because you could take some constructive action.