So communities where people know each other are better but, oh my gosh, don't be polite and get to know your local patrol officer?
At least here in my part of California, we do not have "local patrol officers." There is no community service program where citizens can get to know CHP, local PD/Sheriff or vice versa. There are no beat cops that we can engage in some benign conversation or relay a concern. Any communication with the local PD is only through email (or 911 for emergencies.) Their offices are closed to the public. I understand that CHP patrol officers are supposed to live many, many miles away from their office. This could be for personal and family safety, and I get that.
When we moved into our home 24 years ago, the gentleman across the street was a very high ranking CHP officer--two stars on his collar. When he and his wife welcomed us to the neighborhood, they refused entry to our home for a soft drink or some munchies. His parting words to us were, "Again, welcome to the neighborhood, but do not ever talk to me about the CHP on any matter whatsoever, or ask us to any social event." When I relayed his message to our other neighbors, they all said the same thing.
This sort of isolation from "society" only serves to make police departments more insular. So, NigelM, we tried to get to know our neighbor, the Division Commander. It was he who refused any sort of hospitality and assumed that all of us here in the 'hood were unsavory types who would compromise his job and ask him to commit a misdemeanor by fixing our next CHP-issued traffic citation. Our only CHP traffic citation was in 1981--so 33+years and counting.