Of course, I'd never get rid of my pets.. but if something happened that forced me to rehome for the comfort and safety of my own pet, I have relatives who would take the cats & dogs and friends that could take my rodents & birds. I've talked to them about it at some time or another, and they said they'd want to take them instead of seeing them go to a strange home.
Though, out of the two choices, I'd put a price on my pets... especially the rats(as if some reptile people find cheap rats for sell, they'll use them as food). And like already mentioned, if a person isn't willing to pay a small fee for the animal, most likely, they won't want to pay for vet bills.