I was one for a few years. There were no limitations for transgender officers at the department I was with. The issue some people have run into is that most departments have a lot of things that are determined by sex - restrooms/locker rooms and fitness standards as easy examples.
I worked for a major department on the east coast. We had a male locker room and a female locker room. There were no single use facilities in any police building I ever worked in. So if you're openly trans during academy training, you may have some uncomfortable shower situations. That typically lasts for 4-8 months depending on who you work for.
Fitness standards seemed to trip a few people up at my department in the past, mostly on the FTM end. You'd most likely be expected to meet the standards of the gender you identify with. So for FTMs, they would need to meet male fitness standards. These are often a lot more difficult than the female standards. They're doable, but difficult.
My advice would be to make a list of departments you'd want to work for and contact them directly to get advice about your specific case. Also - not wanting to rain on your parade, but just offering a reality check - despite the climate toward law enforcement in the US right now, law enforcement jobs are still very highly sought after and competition for openings is steep. For my department, 2/3 of my academy class was either prior military or had a Master's/PhD. So getting yourself in great physical shape and getting at least a Bachelor's degree is highly recommended.
