I know what I'd do. I'd build another coop & run and allow the skinny chickens to move in. I'd keep the coop well supplied with feed and fresh water, and tell the neighbors their chickens have moved into my new coop and won't leave -- Do they want to come get them back?
So then the neighbors have to fetch their own chickens and figure out how to prevent them from "free-ranging" back to your coop.
Because you know that if the choice is left to the birds, they'll stay in the home where their needs are met.
I wouldn't tell the neighbors to feed their birds more. I'd just say that their birds love me because I feed them.
As long as they think they're taking advantage of you, those neighbors will continue to do so. But if they start to feel that they way they feed their birds (not enough) is allowing YOU to get the advantage over THEM -- that their chickens are giving YOU their eggs! -- they will need to put a stop to it.
They aren't just "proud" -- they're narcissists. They won't stand for being anything less than top dog... And that includes with their chickens. They will need to prove that their chickens belong to them.
By the way, I know EXACTLY where they got the idea to let chickens "free range" without supplemental food or water -- check out this horrible thread on Survivalistboards.
https://www.survivalistboards.com/threads/youre-doing-chickens-wrong.963280/