I wouldn't want someone I know to start seeing my T. If someone would ask me the name of my T and where she works, I wouldn't tell. I don't think that's selfish. My current T is a very good T and she's a very good T for me. That doesn't have to mean she would also be a good fit for someone else. She's wouldn't be a bad T, but it just doesn't have to mean she would be a good fit with everyone.
I might recomend the practise where I first saw her (she doesn't work there anymore), but I wouldn't tell where she works now.
I know she has many other clients, but I just don't want to share her with someone I know. That might also come from some insecurity.