There would be positive externalities similar to those that would occur if fire fighting were provided by private enterprises. If one person's police force arrests a person who is stealing from or harming him, that criminal will be taken out of circulation and thereby be prevented from harming others. An individual might therefore expect other people's policemen to protect him, thereby receiving benefits for which he is not paying.
But it is also possible that private enterprise police forces could produce negative externalities. Private police forces that are especially strong and competent could, on behalf of those who hire them, exploit individuals whose police protection is weaker. For example, how would private police forces serving the mafia be expected to behave? In a country policed by private enterprise police forces, there would probably have to be created a public police force to police the private police forces.