I think the life about knowing that stuff about her might be more disturbing than actually knowing it, though if she's more guarded than the average t,even you knowing that stuff would be disturbing if she's trying to keep her private life out of the office...
I think it's fairly common for clients to be curious about t's who share nothing about themselves. I think likely even for t's who share moderate amounts about themselves, some clients will feel the need to find out more...
I've looked up t's to a certain degree, though I am cautious to find out too much for fear of it negatively impacting my therapy (I've experienced some pretty distressing dual-relationships).
Current t is relatively open about things if asked. I've actually asked her to refrain from telling me too much since I worry about t if I know potentially distressing info (like minor health issues)...