You seem to be correct in that law enforcement uses 'persons' over 'people' more often. [1] I'm not entirely sure where the distinction comes from, but language can be such a funny thing.
Strange how I'd never noticed "missing persons" as being an odd construct before. I guess I'd just heard it so often I wasn't really aware of how the phrase was constructed.
[1] http://grammarist.com/usage/people-persons/