Well, clients search their Ts, Ts may search their clients. Fair game to me.
Honestly, I think everyone should be aware that whatever we post online can be accessed by anyone. If there's something we don't want to be known, we should post in on our social networks.

I'd trust a good T to do a search using their knowledge and competence. If criminal records popped up, well, that might be something they would like to know.

I wouldn't mind my T internet-searching me, but I'd rather have her ask me what she wants to know because I'd tell her.