Firstly, even if a position is 100% clinical or 100% research, your training will have prepared you to do some small part of the other -- that may have to be on your own time, but you will have the training (and hopefully connections) to do so.
I'm on internship right now, and the impression that I have is that yes, most positions will have a vested interest in you focusing your time and attention on specializing in either clinical work or research. If you're hired as a professor in a psychiatry or psychology dept to do research then they're going to want you to be pumping out research products and grants and having as large an impact on the field as you can. They're not going to be paying you to see patients.
If you're hired as a clinician by a VA or another health care system they're going to want you to be focused on direct, billable, patient contact hours, or at the very least improving their clinical care procedures such that you're helping them increase patient contact hours or the quality of patient care within their facility. They're not going to be paying you to work on grants, conduct research, or produce manuscripts.
There are exceptions to this general rule, but from what I can tell it's basically up to the individual to create space for the other side within a position in one of these two domains. The position isn't going to exist before your advocate for it and provide evidence that it is fiscally sound.