You know, Leia could tell them a lot about herself without giving anything away. And because the events haven’t happened yet, they shouldn’t align with anything they could look up, but Obi-Wan would believe her because Force. Like, I was the adopted heir and daughter of a noble family. I survived the genocide of my people and the destruction of my home, along with my parents. After being made a Princess of ashes I joined the rebellion full time and fought for several years. Then when we thought we’d won, there was a mission my brother and fiancé and all my friends went on. None of them came back, so I left. Then I ended up being pulled out of a sandstorm by Shmi. Why not say something like that?
Well, she could tell them more. But given how much trauma she is dealing with and how much control she is trying to exert over her environment, how focused she is on saving the galaxy as opposed to, oh, making friends, it’s probably not going to happen. Even if there was a reasonable way to do it. (Yes, she does make friends anyway, but she’s very cautious about it.)
Leia does tell people things about herself when she’s worried that if she doesn’t she won’t be able to stay on their good side, to keep helping or getting help from them. And sometimes she slips. But given that when she slips it comes back to bite her (see, Cody telling everyone about her dead brother), she’s not likely to tell people things on purpose. They aren’t paranoid enough, and she doesn’t want anything to get back to Palpatine.
