Could you be more specific about what you mean by " professional " police ? And also what place you speaking about ?

Because, for exemple, what would be considered a " professional police " existed in France in the 17th century. The Edict of the 15th of March 1667 by Louis XIV, creating the office of Lieutenant of Police is often said to be the date of birth of the modern police in France.

It could help you having better answers if you were more specific :)

I believe that most "crimes" and what not was solved in the village or town area. Most villages has a so called "council" where they discuss crimes and discussions about who owns what area etc.