Good question. No, the data is downloaded only once.
The server tells the client what requests it's making on the client's behalf.
If the client's JS tries to make a request for a URL that's already in-progress, the client's React-Server code will skip the request and return a promise of the server's streamed response instead.
If the client's JS tries to make a request for something, and the server IS NOT already handling it, then the client will send out an HTTP request, and React-Server will step out of the way.