I am a clinical psychologist working in a federal correctional complex (we have three prisons in our complex ranging from low to high security), so the simple answer is Yes, a career as a psychologist working in a prison is definitely possible. I can tell you that I've enjoyed my career thoroughly, although it's not without its stressors and challenges. I have a Psy.D. in clinical psychology and did not need any specialty training. Our agency, the Bureau of Prisons, hires clinical and counseling psychologists. At this point, licensure is not required of applicants.
If you have any questions, I'd be happy to answer them. Best of luck in your pursuits...