I think all therapists know we are curious about them. Ask your therapist a few simple questions and see how she responds? Therapy is supposed to be about you so your therapist wants to focus on you, not her. But an occasional question as you are discussing something germane, "Do you have any brothers?" when talking about your brother, for example, (or sister, etc.) would not be too far afield. Just asking a whole bunch of questions all at once might not go over well though :-) The trick is to be able to ask something at the "right" time so it doesn't strike your T as avoiding your own stuff. If you're having trouble talking about your brother and suddenly ask her if she has any, that would probably been seen as trying to get the focus off you/your brother and probably wouldn't fly :-)

T's do say things every now and then on their own though; my favorite was when my T had to call me to say she'd be late one morning because her garage door wouldn't open :-) She was saying how her husband was over at the neighbor's to get their help (about how it worked). So I got a little picture of her life. Her son once called when he was 8-9 and we were in a session and read her a poem he'd written her. She told me that much so I have a slight clue how old she is, etc. :-) I know when she got her PhD and from what school. Another thing she spontaneously told me was when my husband and I were taking sailing lessons and she had gone to the same sailing school (that was really weird, I would never have pictured her taking sailing). It can take a bit of time and getting to know one another though; T's have to get to know and trust you too! They don't know who is going to walk in their front door :-)