Yes and no. One gym I worked out, the clients were allocated to me. However, I've just come out of a short stint at another gym where I was contracted and they weren't (unless it was a promo or something). But yes - some PTs I have met won't stick around unless they are training their clients or themselves.
I would like to know though - if you were doing an exercise you didn't realise was incorrect, would you appreciate a PT showing you the correct way?