Do ERC721 tokens get saved to wallets? Or do they reside only on the Ethereum network?

I've been trying to find the answer to this, but I can't. I'm trying to understand how the ownership part of the token works. Do people have wallets that hold a bunch of ERC721 tokens in them? Or are the references from unique tokens to the wallet address just saved on the Ethereum network, and ownership is verifiable that way.

Submitted December 29, 2018 at 02:08PM }
via reddit