Oh that is too funny!!!!!
We have a sort of similar story. We have a 125lb Golden Retriever (Gizmo) who was IMPERVIOUS to our electronic fence. Just didn't care. Blew threw it no matter what to bolt across the way to go see the dogs who live next door and steal their toys. They HONOR their fence, and would watch Gizmo come over, happily play with him, then watch Giz leave with a toy, bone, shoe of theirs in his mouth. I called the fence company and told them "Turn him into a FRENCH FRY. I don't care. Just make it work." No go. DOGGY GONE a'socializin'. We got a call one morning from our neighbor whose houseguest was missing his shoe. It took him a few minutes but he realized, after not finding it in the yard where HIS dogs would normally steal away with it, that OUR dog must have stolen it. Sure enough, there it was in our backyard. Another time I found a pair of underwear in my garden. Remembering his rather embarrassing puppy habit of running around the house with my bras in his mouth (I swear to God) I thought it might be mine. Nope. They were big, polyester, and beige. In a flash I realized that they must have been my neighbor's. (Unless my husband was fooling around with an old lady). My Mom happened to be with me at the time and suggested I wash them out and return them. A second later, upon reflection, we decided that that would be horrifying for all. That was 3 years ago and I've never said a word. I just gather up whatever Gizmo steals and about once a week return them all to their yard! (Minus the underwear!)