yes, I have an MPH! You would have to go to one of the schools that is used to training physicians as we have different needs from the non-medics and there are only a few of those programs (Harvard, Hopkins, Columbia, UNC-Chapel Hill, UW, Berkeley are the main ones, maybe Michigan and Emory) and have a specific concentration in mind. If you are interested in health services research or psychiatric epidemiology then you will get your basic epidemiological training and a chance to do some research projects. If you are interested in mental health policy and working at the state or national level or influencing organizations then doing health policy would be useful. If you are more interested in management and administration then you would be better off doing an MBA. One of my attendings who also has an MPH (like a lot of psychiatrists and residents at my program) said he wish he had done an MBA instead. Global health is a bit of a non-concentration really. Enviromental health is not really relevant to psychiatry.
I think the naysayers are wrong. there isn't much you would learn at a decent public health school that you would learn at medschool and you can definitely learn some pragmatic stuff that would be useful - e.g. program planning and evaluation, designing surveys, quality improvement stuff, how to evaluate interventions, doing health needs assessments, how to analyze epidemiological datasets, how to develop new psychiatric rating scales, and it is useful to understand how us healthcare policy works etc. It is just there are very few schools that can offer what you might need, and you could probably learn most of this (except maybe statistics or epidemiology) on your own or through practical experience in the field.