I don't ask my T many questions about her (mostly I say "How are you?" and she always says she's good) so I guess I haven't really tested the waters to see how much she would be willing to tell me.
She will sometimes tell me small tidbits about herself when relevant or if she feels it's helpful (for instance I mentioned being a fan of a sports team and she said she was too as she went to school there).
To be honest, there is a part of me that is dying to know every detail about her. I want to know her backstory, if she is married or seeing someone, is she happy? But knowing those things would mean our relationship would no longer be therapeutic, so obviously that can't be. Even chitchat about unimportant life things would be a waste of time to me -- as much as I am interested in her life (because I care about her, not because I'm a stalker or something) I need to spend my limited time in therapy to focus on me.