It's crazy how much browsers are becoming operating systems. I'm neither for nor against it in general; I simply think it is an interesting thing to watch happen. However in this case, I think it makes more sense for the user to download a desktop application in order to get push notifications directly from the server when not browsing the website.
I guess they don't want the app to hold open connections to every site (some people have solved the C1M problem but some haven't) and they probably want a server to buffer messages when the app is not running. You'd run into these same issues whether the notifications are received by Firefox or a different desktop app.
It's a web app, it already has a server. It must be possible to design this in a way that you use only your own server in the process, and not mozilla's.
