I haven't looked into this at all, but for desktop I would expect the browser to maintain a WebSocket connection (for all websites) to the browser vendor's "push server", if WebSockets doesn't work for some reason, then it can fall back to long polling.
For mobile devices it's discouraged to maintain your own TCP connections for power usage reasons. So I would expect browser developers to have their mobile browsers use the mobile's native push service (e.g. for Android or iOS) to register for pushes, then the browser's backend push service sends push notifications via Apple/Google's push servers which in turn send it to mobile devices themselves.
For mobile devices it's discouraged to maintain your own TCP connections for power usage reasons. So I would expect browser developers to have their mobile browsers use the mobile's native push service (e.g. for Android or iOS) to register for pushes, then the browser's backend push service sends push notifications via Apple/Google's push servers which in turn send it to mobile devices themselves.