Very complicated issue and with no set rules.
I don't know much about American Zoos but in Europe nowadays a lot of animals, particularly those endangered ones in managed breeding programmes, are 'pooled' between the Zoos, and money isn't involved when exchanges or transfers take place. I imagine its similar in America. But against that, Zoos still retain ownership of a lot of the animals they have, particularly where they are, or are descended from stock, that Zoo purchased in the past, e.g. from animal dealers or another zoo. With these, when exchanges take place, that ownership can still stand. If you look at e.g, the Gorilla studbook, you will see original ownership for all animals is still listed although an animal may move zoos several times in its lifetime. Most species studbooks have similar 'ownership' and 'loan to' categories in the animal movements/dates section of the data. Sometimes an animal is born in one zoo but ownership is to another zoo e.g. that provided a parent animal. This is often recorded too in a studbook on the day the animal is born.
Some places such as Private Collections still hold a more complete ownership of their animals too and may occassionally trade them, involving considerable sums of money if animals are moved to a new location.