It depends upon the police themselves as well.
My supply sergeant at Fort Knox was getting threatening phone calls to the supply room from her estranged husband in another unit. I took her down to the MP station, where the MP Investigators REFUSED to arrest him.
We had a case like that here.
The "Dewberrys" were the neighbors from hell. They lived in trashy surroundings, stole, threatened people, actually shot at people (but never hit anyone), and so on. The standard response by the Sheriff to complaints was, "If we didn't see him do it, there's nothing we can do about it." That, of course, is a lie -- they could direct the complaintants to the judge, have them swear out a warrant, and make an arrest.
The Dewberrys also kept vicious dogs which killed sheep. The reponse to complaints about that was, "That's a civil matter. You'll have to sue them." That's also a lie -- the Sheriff can deal with such dogs. And as for sueing them, you could never collect.
Finally, the Dewberry's dogs were killing some sheep. The father of the girl who owned the sheep came out and shot the dogs with a .30-30 (Perfectly legal in Arkansas -- the dogs were depredating on his property.) This occurred near the property line, and Dewberry came roaring out of his house with his harridan of a wife behind him, urging him on. He fired at the father with a .44 Magnum. The father returned the favor with his .30-30.
Dewberry was killed. The bullet passed through him and paralized his wife.
Then came the denoumont -- Both "Dewberrys" were federal fugitives, him for kidnapping, her for drug dealing. If the Sheriff had acted on the first complaints, they'd have been fingerprinted, identified, and carted off to prison.
The Sheriff saw the handwriting on the wall and did not run for re-election.