Agreed - as nice as free ranging can be, there are risks. If the neighbors' dogs don't get your chickens, there is always the threat of hawks, coyotes, fox, mink, coons..... I'd put up a fence to keep them out of the garbage, away from the chickens and away from whatever else they are getting into. The neighbor really has no respect for you if he won't keep his dogs off your part of the property and refuses to take responsibility for whatever mischief they're getting into.Ditto that^^
It's tough situation, with friends involved something like this can break down a friendship quickly when there's a disagreement on responsibility.
You might not be able to convince them to change their outlook and behaviors, but you can control your own behavior.....and in the long run it is your responsibility to protect your chickens whether it's the way you want to do it or not.
You already have a secure run...maybe think about some portable electric poultry netting to protect your chooks while they 'free range'.
It's not cheap, no fencing is, but it is something that you can take with you if you move to another location and is very effective at deterring predators.