Does this work cross-browser?

[KosGrillis]

Can chrome send data to firefox for example?

[malcolmocean]

from stackoverflow:

Local Storage is "local" in that exact browser and ONLY in that browser. To retrieve something stored in Local Storage, you must use the same browser, the same key and retrieve it from a page in the same origin (e.g. domain).

If you want something available across multiple browsers, then you have to identify the user somehow (usually a user login) and then you need to store the data on a server somewhere so that the data can be served to the same user across multiple browsers.

[qm3ster]

If both are open, you can set up a peer to peer connection over WebRTC.
You still need a server for signaling (establishing and maintaining a connection) but you can reduce the load of connections to your server (and latency) vs just syncing through the server.

This also works to P2P connect devices over local network.