Constables, but they filled a different role. The burden of filing a case fell upon the individual who had been wronged, not a detective or a prosecutor, and the only likely intervention from a constable was corralling a defendent who didn't want to appear in court.

You should also keep in mind that most police forces were not formed to do "police work", but rather to implement control of unruly crowds or certain social groups, usually immigrants or free black communities. I assume the formation of the Boston Police Department was almost a direct response to the change in the social order caused by increased European (especially Irish) immigration. I thought the earliest police force in the United States had started in Charleston in the early 1820s in response to a growing free black community and the discovery of a attempted slave insurrection.