Google Chrome Push Notifications not working if the browser is closed?
According to Can I Use, Chrome and Firefox desktop browsers require the browser to be running for receiving push notifications; mobile browsers typically don't.
On desktop, browsers need a process running. For example on Mac OS X the browser can have no window open but if you look at the dock, the light underneath icon can be glowing (meaning it has a process running). In the scenario you should receive push messages.
If the browser was completely quit, then push messages won't get through.
The same applies to windows and Linux.
On android you should be receiving the messages regardless of whether the browser is open or not. This is in part (as far as I know) to the fact that android manages it's connection to the push service rather than the browser, so it'll receive messages whenever possible.
The messages should (eventually) get through even if they are sent while the browser is closed, or the device is offline, etc. (And this works for me using https://gauntface.github.io/simple-push-demo/ and other tests.)