This really isn't advice, and I'm no anthropologist. Just some things I have observed watching my daughter develop and interact with other people.
I think we often forget that we are primates, and have primate impulses, especially in children. As parents we have to "tame or civilize the primate" in our children.
In your situation, the kids next door sound exactly like a troup of chimpanzees or baboons, taking whatever they want, breaking or just dropping when they tire of it. They don't care about the item at all, just that they got it from someone less dominate than they (hmm, maybe the neighbor boys are trying to show their dominance over your kids).
I'd only let the kids play outside in your yard with a few old toys. Don't let them into your house, and don't feel bad about telling them that you don't trust them in your house. They need to learn that trust has to be earned, and they have proven they don't deserve it. If they need to use the bathroom or get a drink, their house is just as close, as another poster stated. Would you let a troup of chimpanzees in you house?
Where is their dad? Can you talk to him? Obviously, their mother doesn't care and is probably following the mantra of "Boys will be boys." (to their detriment) because she doesn't want to deal with it or is trying to shift the blame and their problems to you. Don't let her get away with it! She has to raise and discipline them, not you, and their behavior reflects on her good or bad.
Talk to your boys about their toys. Try to explain that the toys cost money and you are not buying gifts for the neighbors. Do your boys get an allowance? Maybe it's time for one and all toys are bought with allowance money. Perhaps then they will be less likely to let the chimpanzees get ahold of their toys. I was also thinking that your boys like to show off their toys as a bid for dominance, and then the chimps have to reassert their dominance by taking the toys from the boys.
Try to expose your boys to other kids, but don't totally cut off they neighbor chimpanzees, or they may become the forbidden fruit. Just keep them outside where chimps belong.
I'll probably get a lot of heat for this comparison, but it is an interesting way to look at the situation.
I came up with this idea of the untamed primate when I had to deal with an incident in which my daughter's preschool teacher threw away her snack. Would you take food from a wild primate? Well, her teacher got what she deserved in my opinion, but of course my daughter had to learn that civilized people don't act the way she did.