My T is pretty open about his personal life- I do have to ask though.
I did google him in the beginning- his professional life. I guess I'm not interested in more- could be because he's so open. I would not care much for a blank slate T.
I do still believe that to google your T is pretty much OK, somehow I don't think so about his/her family/kids esp- but that's my bias, weird I know.
I guess if you want to tell him do so- why not? Therapy is a place to be as open as you wish.