[CFrance]

I like Patty55's idea. Just show the neighborhood that your dogs are well-behaved. I wouldn't try apologizing either. You already did once, and they didn't accept it. Give the neighbor lots of space, and things may change later on. I'd give your direct next-door neighbor a wide berth too! They shouldn't have related the "troublemaker" accusation to you and make you feel even worse.

Dogs do get out every once in a while. We've had two wander into our backyard and were able to reunite them with their owners. A year or so ago our golden also got out the same way yours did, ran three yards down and into our neighbor's garage. Fortunately, she loves dogs. It's pretty embarrassing when you go out of your way to ensure your dog's not causing a neighborhood problem, and then they slither under your radar.

Goldens just want to say Hi. Your neighbor over-reacted.