The answer to this question depends a whole lot on your definitions of "private practice" and "research."
"Private practice" is a tricky term. For instance, Fox Chase cancer center isn't directly paired with a University and (I believe) is for profit. They run a number of cancer trials, some run by pharma, some not. Is participating in, say, a Genentech or Amgen phase II or III cancer trial "research" by your definition? If so, then yes, absolutely. Are you going to get authorship? Maybe, but probably not - but why would you care if you're not in a tenure track job?
If "private practice" is you hanging a shingle, or operating out of a smaller practice group, you probably won't have the protected *time* or money to conduct meaningful research (what a University provides). But again, I've seen small groups like these helping enroll patients in trials run by someone else. You won't be running your own trials, because this is expensive and requires a lot of infrastructure (for starters, whose IRB will you go through? Who's paying for the labs?). Basic science research is out of the question.
To another poster, no, your research doesn't have to be in the purview of your specialty, but it ought to be peripherally related (simply for you to have face validity at journals and among colleagues, and when applying for NIH grants).
Getting down to brass tacks OP, is this a serious question, or are you just asking because you plan on a private practice anesthesia or derm position, but want to tell schools you're applying to that you have research aspirations? If so, it's a pretty transparent argument. Or, if you're worried about being able to see patients AND do research, then clinical research is the place for you. You'll make a little less money, but an academic clinical researcher often operates on a 20% clinical/80% research split, so you're still seeing patients. I used to think it was a false dichotomy of all research or all academics until I started seeing how these things actually work. Consider MD programs with a strong Epidemiology footing.