It's up to others to decide whether things like Eugene's CAHOOTS program is a contributing factor to Eugene's very high crime rate or if it mitigates what would otherwise be an even higher crime rate.1 Instead, let's talk about a world in which police forces were greatly reduced. What models have existed in the past and might be used if the police became a minimal force?
Constables: The 1,000 Man Model
The Idea: I was doing some research about constables recently and depending upon what period of time you research you find them having a variety of different responsibilities. At one point, it appears that constables were to be in charge of 100 men (basically the head of a household) and had the responsibility of making sure every man was armed (so he could be called up in levy) and that this group enforced the law in their zone.2
How It Works: An adaptation of this model for the modern world would have to include many more people per constable. Perhaps 1 per 1,000 adults would be a more appropriate. So, New York City, with a population of 8,399,000, would go from having 36,000 officers to 8,399 constables. If we take out those under 18 and above 55 it should lower the number to approximately 6,000.
In this system the constable would make certain that the adults in her area were armed well enough to respond to criminal activity. That would probably mean at least one rifle per adult and ammunition. When a criminal act occurred the constable would call up an appropriate number of these adults who would show up armed and act to take the culprit into custody.
Difficulties: From what little I've read on this, problems back in the day used to rise when a person committed a crime in a town and then went back to his own area because the members of the group in the area he lived in neither felt the need to arrest someone who was living peaceably in their area nor felt inclined to let members of another group come into their area to apprehend the criminal. I'll leave the exact parameters of that issue to anyone out there who wants to do some historical research.
Other issues that would seem probable to arise would be the difficulty of getting people to react quickly enough to respond to significant crimes such as robberies or kidnappings, the free rider problem in which a number of people won't respond because of inconvenience or because they perceive the danger to be too great, and the lack of training of the individuals responding which can make them cause much more damage and harm.
Posse Model
The Idea: This is a modification of the idea above wherein there is a law enforcement officer or two for basic peacekeeping and when a major crime happens he summons citizens generally to chase down the bad guys. If you've seen a western movie you've seen this in action.
How It Works: This is a less organized version of the 1,000 Man Model. The lawman doesn't organize the people in his area or make sure they are armed. He just calls them out when he needs them and assumes they will come and come armed and with ammunition.
There are actually laws still on the books from this model. In Virginia there are two statutes which are artifacts of this system:
§ 18.2-463. Refusal to aid officer in execution of his office.
If any person on being required by any sheriff or other officer refuse or neglect to assist him: (1) in the execution of his office in a criminal case, (2) in the preservation of the peace, (3) in the apprehending or securing of any person for a breach of the peace, or (4) in any case of escape or rescue, he shall be guilty of a Class 2 misdemeanor.
§ 18.2-464. Failure to obey order of conservator of the peace.
If any person, being required by a conservator of the peace on view of a breach of the peace or other offense to bring before him the offender, refuse or neglect to obey the conservator of the peace, he shall be guilty of a Class 2 misdemeanor; and if the conservator of the peace declare himself or be known to be such to the person so refusing or neglecting, ignorance of his office shall not be pleaded as an excuse.
The first authorizes a lawman to order citizens to help him. The second allows judges and prosecutors to order citizens to go get the criminals and bring them back. Of course, neither is much used in the current era of professional police forces, but they're still on the books.
Difficulties: This has much the same problems of the 1,000 Man Model. It would be difficult getting people to react quickly enough to respond to significant crimes such as robberies or kidnappings. There will be a free rider problem in which a number of people won't respond because of inconvenience or because they perceive the danger to be too great. The lack of training of the individuals responding could make them cause much more damage and harm. Beyond these problems is the probability that without some sort of supervision to make sure they even have firearms many individuals might not have a suitable weapon or adequate ammunition to accomplish the job if they are called upon.
Vigilance Committees
The Idea: These are committees of concerned local citizens who come together to decide who the troublemakers are and how to handle them. These have played a part in US history and in some places are looked upon with a sense of nostalgia. Montana even has the identifying logo of one of these committees on its highway patrols' patch (3-7-77).
How It Works: Historically, these were groups that came together privately, made their decisions as to whom the bad people in their communities were, and then acted as a group to deal with the bad people. In some cases this meant immediately going after the individual as a group, but in some cases (the 3-7-77 group listed above) the group would first give a warning so the bad person could clear town. In the end, the committee would apply violence as a group to affect a solution to the problem. This could have been anything from riding the bad man out of town on a rail, tar and feathering him, or just out and out killing him.
Difficulties: Some people look upon these with a bit of nostalgia, but they are incredibly problematic. One may be a necessity if there are no available lawmen, but it essentially makes the committee both jury and enforcer, thus eliminating any meaningful due process. Additionally, there is no guarantee that the committee will represent a majority or even a large portion of the population. An armed committee can impose its will thru fear and firepower even if it is a small minority. This is the realm of know-nothings and klansmen. If any form of organized law enforcement checked by courts is available this it will almost always be superior to this.
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These are the primary other historical models I've been able to find, although I guess Mob Justice could be another. Personally, I view mob behavior as perhaps vengeance, but not justice or policing. Good luck on choosing a replacement for lawmen.
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1 This strikes me as an extremely difficult metric to compose and thus will be a matter of political and social philosophy unless and until there is some sort of a decision based on a serious lack of fiscal resources.
2 Yes, I'm sure this and the other models offered today are filled with all sorts of historical inaccuracies and lack a nuanced understanding of how things actually worked. I'm talking about models here, not trying to get a PhD in obscure historical matters.