Normally, a professional interpreter will limit interaction with a client before and after assignments. Other than a pleasant introduction and establishing language preference there really shouldn't be more conversation prior to the assignment. At the end of the assignment it's acceptable to say a pleasant good-bye but even that's not required, especially if the Deaf client doesn't appear to be in the mood for pleasantries.
Same goes for the hearing client of the assignment. Interpreters shouldn't chit-chat with the hearing parties either, other than normal greetings and any issues that would impact communication. Those issues would be determining if the hearing client knows how to use an interpreter, seating and lighting, any specialized jargon that might be used, or name references that might be used during the assignment.
Obviously, everyone involved is human, and if you use the same interpreter year after year, it's possible to develop a friendly relationship. The interpreter needs to be careful to let the client take the lead in how much to share, and even then, keep things professional when "on the clock."
One thing interpreters are taught is that we shouldn't show up to an assignment too soon. (Obviously, don't show up late either.) The reason is, that limits advance contact with clients. If it appears that the terp will be left alone for more than a minute or two with either client, then the terp should excuse himself/herself and wait out in the hall or waiting room until the other party arrives. That takes off some of the pressure to fill the waiting time void with conversation.