I can tell you this. I wasn't a MP, but I was in a rifle platoon. Either one you do will be fine. I can't speak for the rest of the Armed Forces, but the good Ole University of Science Music and Culture's MP's love their jobs. Yes, you will most likely go to Iraq and most likely ride convoy security, but you'll do that with a lot of MOS's.
As far as working for a Civilian P.D., I can tell you one thing. Most people don't like it because of the image people have, but Private Security Services most of the time pay better than a local P.D. It's true that you don't have the retirement system that they do, but you also don't normally have to work 12 hours shifts (depending on location), don't have to deal with these stupid people in the public, or have to go to court and mess with these people. It can also lead to other job opportunities in the Industrial Security field that pays big bucks. And, the good thing is that most private security companies will eat you up for having MP or Combat Arms experience. These are just my opinions. I have worked with the public as a Professional Fire Fighter, and been around the general public with the F.D. and Rescue for most of my life, so working with them is rewarding, but frustrating as it can be most of the time.