I know of psychiatrists doing research and maintaining a small clinical practice (about half-time each, from my understanding) -- they work for a large, New England university, in an affiliated research and treatment center. So, yeah, the opportunity is there, and this cannot be the only place offering such a set-up. I am sure that Dr. Polymath is right, and would also guess that most physician-researchers are probably NOT holders of both degrees.
In medical school, at least early on, your ability to do research might be a bit curtailed by the obscene amount of classwork you have. However, I have classmates who do research with our professors. Some of them have fewer classes to take because of prior academic work, some of them are just wicked brilliant bubbas, and some of them have no life outside of school.
Some medical schools will DEFINITELY offer more research opportunities than others, so it is good to make sure that you are an educated consumer in that regard! Also, some schools offer more research opportunities in some specific areas, and next-to-none in others. For example, at my school there are opportunities in pharmacology and diabetes and heart disease, but not much available in public health or preventive medicine.
I think that, unless there is a specific reason for you to want to add a master's degree to your premed resume, you will probably get plenty of excellent training in residency for whatever you want to do. That being said, I earned a master's degree before medical school, but it wasn't until grad school that I really figured out that I wanted to become a doc.
If you look at the articles in medical journals, there are plenty of researchers doing publishable work with just their medical degree. Sure, dual-degrees are cool, and they add letters to your nametag, (and, heck, in Germany they would actually call you Doctor-Doctor McGillicuddy, acknowledging both degrees), but it isn't an absolute requirement for the career path you have outlined. Remember that lots of psychiatrists don't have undergrad or graduate degrees in psychology at all... they went through med school, and then completed a residency in psychiatry. That seems to get the job done!
I would not try to talk someone out of a dual-degree program if that is what they want to do, but I would say that one should always be sure that they know all the angles first.
Finally, while Drs. Admin and Polymath disagreed with the reasons behind my idea of taking A&P before medical school, I do have to admit that I really-really-really enjoyed the classes I took instead of A&P... they were literature classes. (Maybe we didn't disagree that much after all!)
