Right now, by virtue of finishing USMLE Step 1, you're eligible for a medical license, and can practice medicine at an urgent care/doc in the box setting. The ACGME is lobbying to make changes in the Prev Med curriculum. Their position is that the MPH year isn't really a year of graduate medical education and some would like to see that year replaced by more clinical rotations. Most prev med docs find their MPH training invaluable and critical to their public health practice, and feel like the quality of their practice would be compromised.
I'm in the military and only trained in Prev Med, but I also practiced for 8 years before going back to residency. Prior to going back, I had a couple offers to moonlight and cover practices in the above settings, and probably could still do so. Hence, you could probably see patients in a group setting without primary care training, but that would depend on the job and the practice/insurer requirements.
I"m not aware of any limitations on only Prev Med trained physicians, but I do recall that one of the grads from my program (UNC) received a faculty appointment in another state a few years after graduation and had some trouble getting a medical license in the new state because she couldn't prove her clinical competency (hadn't seen patients in a while..)
Currently, (when not deployed) I see patients one morning a week in a primary care setting, but spend most of my time providing consultation services (for Navy and Marine Corps units) and research (norovirus and h1n1). I imagine when I get out of the military I'll do a combination of clinical work and research/policy work, (many former military prev med docs do research or consulting)
Hope that helps.