I thought those were called “rescues” not shelters. I’ve heard of the places that are privately owned and accept donations. I’ve never actually seen one. But I know a lady that used to volunteer at one.
I lived in a rural area that had a shelter that served and was supported by the township. There was a volunteer organization that raised money to help families spay/neuter their pets. I suspect that city shelters are affiliated with the Humane Society.