Contrary what some have posted about not needing a high school diploma and basically they hire anyone, they're wrong. It's a very competitive and tough to get hired. It depends on where you live, but sometimes there are 500 to 1000 applicants applying for a few positions. Where I live, the cops make pretty good money ($60,000 starting and top out around $100,000) and great benefits that include a pension after 20 to 25 years. They require a minimum of 60 credit hours, but many have a 4 year criminal justice degree. You're not going to be able to have a pro fighting career with most departments. Most departments have to approve all off duty employment. There is no problem with training and it would benefit you on the street. They do like former military. Maybe you could join the reserves and go to military police school to enhance your chances of getting hired.
Like another poster stated, "How good are you at MMA fighting?" If you're not top notch, I wouldn't try togo pro because it's not going to pay the bills like a law enforcement career.
Good Luck