Pretty awful actually.
You claim that it would be far more expensive for private entities to provide security and to back up your claim you mention how these private security firms would be forced to provide protection for non-customers, isn't logistically feasible, and because foreigners might be the producers of these domestic security services. None of these objections hold any weight though. Maybe it's your lack of imagination or maybe you just don't understand how powerful the market is but there is no reason why in principle firms would have to service non-customers and there is no inherent reason why providing security services needs to be a logistical nightmare (you give no reasons why this MUST be the case). Your last objection is just silly once one realizes that we import all types of goods like food for example that can be argued are just as important as security.
If we create a model of the US but assume that no police exists then I think it is safe to assume that there will exist a high demand for protection and security services. Entrepreneurs being what they are would probably step in to fill this void (profit motive). Now the protection and security services they provide would more than likely look nothing like what the current police provides. Furthermore, depending on the area (rural, suburban, urban), the demand for security and protection would differ dramatically. Maybe in the richer pockets of the US minimal security is required and conversely, in the poorer pockets of the US more security is required. I don't see why this would ipso facto create a logistical nightmare for security firms or why if security firm A provides security services for X why must he also provide it for Y if Y isn't a customer.
To reiterate, a society with completely private security services would more than likely look nothing like the current state of affairs. You wouldn't have a monopoly providing services for millions of individuals but more like hundreds if not thousands of firms providing security services to millions of individuals. These firms need not coordinate with each other for this to work either (though there probably would exist a strong incentive to do so). For example just look at what security companies like ADT do today. That is just one aspect of security but they compete with other firms without any logistical problems or without providing security to non-customers. Moreover, because monopolies are by definition inefficient and wasteful, individuals would probably get much better security for less than what it costs them today.