I can't speak to the nineteenth to twentieth century US, which might be what you are looking for. I can, however, speak to early modern England and British America in the sixteenth to eighteenth centuries.

First of all, there was no professional police force in Britain or the colonies for the early modern period. To have a "police" in the modern sense would have translated into a standing army in the seventeenth and eighteenth centuries. A police force, as we understand it today, is a group of men (and women) who serve the state directly; they are armed and trained to keep up arms; their duty is to maintain the peace and guard prisoners. To an eighteenth century Englishman, a "police" looked awfully similar to an occupying army.

What England and the colonies did have was a legal and neighborly requirement to assist in apprehending "suspects". People who were the victims of assault, robbery, burglary, rape, etc. were expected to "raise hue and cry." Raising hue and cry was the early modern equivalent of "calling the cops". People who heard the call and cry were expected to stop any suspicious persons running away, or to find the person calling out to assist.

What did "hue and cry" consist of? It could be as simple as "Stop, thief!" and people were expected to grab and apprehend the "suspect." "Hue and cry" was a critical element in trials for rape, more than any property or other violent crime. That was because women (already under incredible scrutiny and shame) were questioned during trials to ensure that they had made "hue and cry" during the incident. If they had not cried out, the testimony of the victim was criticized, and unless she had firm reasoning why she could not cry out (eg, she was gagged or feared that she would be killed) the accused was more likely to be found not guilty, or have his conviction & sentence reduced.

The idea of "hue and cry" being sufficient to enforce the law was rooted in the concept of neighborliness. People within a city, town, village, etc. were expected to get along, maintain the peace among themselves, and help each other in need. In part, they were charged with keeping the peace, as distinct from the sheriff who kept the King's peace. The former was about maintaining cordial relations, keeping each other in charity (reinforced by the mass/communion), and helping prevent foreigners or outsiders from breaking the peace. The latter were charged with preventing riot, murder, and rebellion more than anything else.