With reference to my earlier post, they just had to see you doing something wrong. No need to chase as they knew who you were.
Benefits/downsides of small town living at a time when cops were local. AFAIK, this is deliberately avoided now. Upsides is privacy, downsides is an us and them divide.
Like yourself, this is only personal experience, no statistical evidence, but, one of my drinking pals is a retired Police Officer. In the sort of time period you refer to, he was a beat Policeman in the Town in which he lived (and still lives). Even now, 20 years after retirement, I have been out and about with him, when he encounters the odd 'petty criminal" from the "old days". Like him, they are now retired, in their case, from their fighting/drunkeness/burgalry activities, the encounters (so far) have been "amicable".