The files would be on Twitch’s CDN.
And then the characters can be loaded from where ever.
So the characters can be localized to the streamer. (I run an extension that has data uploaded from the game to the extension so the game data is local to the streamer, theres no transfer of game data between casters).
TLDR: There are many ways for data to be specific to the streamer/channel and be non transferrable