migtig
aka Mrs. Giant
My previous dog and myself lived in a house that had a fenced in yard that bordered another person's fenced in yard and he had huge German Shepherd's in comparison to my then dog.
My dog was very well behaved, well trained, was one of those dogs I could walk without a leash. He had gone to obedience school, trained in therapy, was just a fabulous dog...until he was in the back yard with the other dogs in their back yard. And those dogs would run up and down the fence barking and chasing each other, barking and chasing each other. Number one, I was embarrassed, and number two I was scared that those huge dogs would get ticked off and jump the fence and eat my dog.
Until the day they did jump the fence and I had four dogs in my yard playing and still running up and down the fence barking at each other. My neighbor and I became pretty good friends, and we discovered that they just wanted some doggie time and to play with each other.
So when we moved to another house with another fenced yard with another dog on the back side in his fence, yet once again, running up and down the fence barking, until my old dog would be too tired and would lay down for a rest, only to start back up as soon as he could.
Point to my story, maybe I'm the odd ball here, but I say, introduce yourself to your neighbors, invite the dog to play with your dogs in "neutral" territory. At the same time you can explain to the kids about not trespassing and respect of your dogs, and possibly give them some pointers on care and training of their dog. Also if you get to know the dog as a pup, you can help command some authority. Socializing your animals is not a bad thing. You say your dogs are up to date on all their shots, well trained, etc, so why worry and not introduce them to a new doggie pal?
My dog was very well behaved, well trained, was one of those dogs I could walk without a leash. He had gone to obedience school, trained in therapy, was just a fabulous dog...until he was in the back yard with the other dogs in their back yard. And those dogs would run up and down the fence barking and chasing each other, barking and chasing each other. Number one, I was embarrassed, and number two I was scared that those huge dogs would get ticked off and jump the fence and eat my dog.
Until the day they did jump the fence and I had four dogs in my yard playing and still running up and down the fence barking at each other. My neighbor and I became pretty good friends, and we discovered that they just wanted some doggie time and to play with each other. So when we moved to another house with another fenced yard with another dog on the back side in his fence, yet once again, running up and down the fence barking, until my old dog would be too tired and would lay down for a rest, only to start back up as soon as he could.
Point to my story, maybe I'm the odd ball here, but I say, introduce yourself to your neighbors, invite the dog to play with your dogs in "neutral" territory. At the same time you can explain to the kids about not trespassing and respect of your dogs, and possibly give them some pointers on care and training of their dog. Also if you get to know the dog as a pup, you can help command some authority. Socializing your animals is not a bad thing. You say your dogs are up to date on all their shots, well trained, etc, so why worry and not introduce them to a new doggie pal?
Obviously they didn't want it anymore AND not only do you throw it away, you throw it in the trashcan with the picked up dog poo! No one wants to dig through that! 
Lose a few toys and I bet the parents start getting REALLY picky about them not leaving their crap in your yard anymore!
But then again I don't leave my dogs outside unattended for extended periods of time, and they will listen to me whe I tell em to shut the heck up!