There really is no excuse for your dog to run at large. My dogs (I have 4) have never been in my neighbor's yards. Although trained well, I leash them even when I transfer them from the car to the house. I crate them when I am not home in a locked bedroom. That way, none of the kids can accidently let them out or make a bad decision. My dogs spend a lot of time in my fenced back yard, but NEVER unsupervised.
If my neighbors told me that they would shoot my dogs if they saw them running at large, it would not worry me. This never happens. We have many loose dogs in our neighborhood and I deeply resent them. However, none of them are mine.
Just do what you need to do to keep your dog safe and contained. It'sll be okay.