Yes, get to know that neighbour, sounds like a good person.
I own a dog too, and i'm not making excuses for animals here, but they're instictive, they can't reason, they don't look at your pool and think "maybe I should ask permission". However, the owner should keep it leashed if it's that rampant. Then again, your gate should be closed......can't really blame anybody here, sounds like something that happened because of a number of circumstances adding up.....
I too own a larger dog that keeps the rodent and pet population to a minimum around our area (at least on my land it does), last night we had a good 5 minutes of entertainment as my dog "played" with a squirrel, the thing would try to outmaneuvre our dog, but he's so fast, he's on top of it continually, he kept it in a 10x10 area, never letting it get away, once i yelled at him, he knew I was getting mad at him and playtime was over, so one final chomp and game over for the squirrel.....
My dog was pals with a jack Russell from down the road, until I caught the thing relieving itself on one of my baby pine trees, since then,I taught my dog to "play" with the JR the same way he "plays" with squirrels, don't see him much these days.......