In fairness, police also act as "first responders", much like firemen. To fill this role, the people of the community would need to take steps to train everyone to act as a first responder able to deal with a wide variety of circumstances from car crashes to heart attacks. This training would also need to address how authority can be asserted, and people can be organized in the event of an emergency. This is nearly identical to how a milita would form in a community to protect itself from an external threat.
There ARE already private EMAs, ambulances, etc. that is, groups which do some of those things and are not a "public" or state service (and so then obviously would not be effected in a negative way by the state being gone). In fact, any of the things an individual imagines "government" to be responsible for, are only due to the actions of the individuals who engaged in those activities because they would be getting a pay check. All of the "services 'provided'" by the "government" can be provided by any group of individuals (other than enslaving everyone, the main 'government' program, anyway).
There doesn't need to be any "authority," (and there isnt o.O) any group with the "right to issue orders" or the "right to settle disputes." (YOU ARE THE ONLY "AUTHORITY.") The dispute is plainly evident. The answer is as simple as the answer to "what do you do if someone invades your home" or "your neighbors home" - attacks you/your friend/anyone else / steals your car, tv, xbox, etc.
nothing about the "police force" is "fair," theyre driving around in cars paid for with extorted money!
"This training would also need to address how authority can be asserted, and people can be organized in the event of an emergency." not sure what kind of anarchist would say something like that :( "asserting 'authority'" = telling you what to do and exercising "control" or "power..." over you.. clubbing you over the head for non-compliance..because it is "right" that they order you around, and it is your duty to obey.. as it is perceived to be their "RIGHT".. they have "settled your dispute," for you, whether you wanted them to or not...