Someone I know just had a community cat of hers caught and sent to the AVA. Fortunately, she was able to retrieve the cat in time before it was killed, but it turns out that a neighbour trapped the cat while she was away. She was upset and aggrieved, that a neighbour would do so before speaking with her as the neighbour knew the cat was a cat she cared for.
It is extremely upsetting to lose your cat in such an upsetting fashion, and it is true that some neighbours are very trigger happy. They would rather trap your cat (knowing it is under your care), rather than wait to speak with you.
However it is a good idea to try and get on good terms with your neighbours BEFORE there is any friction. You don't want your first contact with your neighbours to be one where there is already some unhappiness over a cat, for example.
If you first move into a neighbourhood, or there is a new neighbour who moves in, do take the initiative to go over and introduce yourself. It can be as simple as just offering to give them your number or telling them to knock on your door if they need anything. I tend to like to bring over something to welcome them - baked goods (if you bake), flowers, etc. It doesn't have to be something expensive. That way your start off your first interaction on a positive note. Your neighbour is then more likely to approach you if they do have an issue in the future.
Especially if the cats are very obvious - ie lounging around and impossible to miss - you may want to also let your neighbours know, that you help to care for them. You should also let them know that if there is any issue, they should not hesitate to contact you.
Of course, it doesn't mean that the relationship will not go sour in future no matter what you do. However, it's always better to have a friendly first interaction then to meet your neighbour only when they have an issue with the cats.
A caregiver I know is so friendly and helpful to the neighbours that one of her neighbours who really doesn't like the cats at all, keeps an eye on the cats when the caregiver is on holiday. The neighbour still doesn't like the cats - but she DOES like the caregiver, and that makes all the difference.