AFAIK, React does claim to solve "the view problem" and not more than that. You still need an architecture around your application if you're building something larger than a demo.
Quote from the React landing page:
> Design simple views for each state in your application
Quite literally, one of the selling points of React is that it's focused on the view based on data, not on where that data comes from.
You should be persisting state in the Data layer on the client
The Data layer should be syncing with the servers or peer to peer once in a while