Does your neighbor feed your dog treats?   

I think it is very sweet that they do that. I don't think that it is hurting anything that they do that. It sounds like they enjoy doing it, and it gives them and the dog something to look forward to each day. I don't think the dog is begging, he is just letting him know he is there and ready to receive what he is used to receiving. Af this point it would a difficult habit for your dog to break. This is just my opinion but I would just let things stay as they are. It is not truly hurting anyone, and probably makes the neighbors happy.

What did you say to them and how did they respond?
My dog is big and very friendly, so everyone likes him and they want to give him treats. I tell everyone he's on a diet, he is a large dog and has to be kept in shape. Otherwise it hurts his legs. If someone disobeyed my wishes it'd be serious business because they'd be harming my dog.
Also, all those treats have wheat in them and a large percentage of dogs are allergic to wheat.
If the neighbors know it's not just giving the dog a treat, there are other issues at play, maybe they'd stop doing it.
As far as the begging goes, I'd be mighty pi$$ed off. I believe we, as owners, have the complete and total right to decide what behaviors our animals learn.

As far as I know my neighbors do not give the dogs treats. We live in the country though and the nearest neighbors are not all that close, so if they are I would not know about it.

If you really have a problem with it and you have spoken to them about it and they keep doing it I would mention it to them again. Does the begging carry over into the house or is it only an at the fence thing?

If you're concerned about the dog's health, maybe you could provide them a bag of healthy treats to give to her instead of what they're giving.

When we're at my FILs house, they feed my dog constantly (only approved items, though), so I reduce her food accordingly. When they realized that she wasn't getting as much dinner because of what they were feeding her, they stopped treating her as often.

You could also keep your dog away from the fence so that she can't beg for the treats.

i dont see anything wrong with it. your neighbor are just being nice. i think your dog likes them a lot now :)

If your dog is putting on too much weight, then you will have to speak to the neighbor or walk them more. Otherwise, I guess you just have to live with it. If it annoys the neighbors, then I guess they should stop reinforcing it. :)

There are ways to quietly dissuade such begging. Special collars that trigger a small shock. You can set up an invisible fence to deter your dog from getting to that side of the yard or there are also hand held remote devices where you can just give the dog a small shock when you hear or notice begging behavior.

Both are costly options, so if this bothers you that much, it might be all well and good to simply ask your neighbor to stop again. Politely of course.

They have fed my dogs scraps before but nothing like dog treats and such. As long as they didn't feed her any bones I was fine. I have had animals die because they fed them bones. Small dogs aren't meant to eat certain types of bones. If they wanted to buy the dog treats I really wouldn't have a problem because that is just their way of saying they like the dog if they didn't enjoy it they wouldn't feed her the treats.