Earlier quoted context omitted.
In my experience, they usually won't tell you who the clients are unless they are actually interested in you for the role. I have tried this with one recruiter, and it didn't work well.
If they don't tell you the client then why would you continue interacting with them?
I find it slightly annoying if they don't give those details up front. If they refuse to give them when asked, we have nothing more to talk about.