Technically, webpush-enabled site starts a service on your phone (read on js service workers) that then asks a browser to accept notifications from some origin and sends permission credentials to that origin. Then it decides for every incoming notification if it is worth popping up (e.g. if a site is a chat and it is already on the screen, there is no need to spam with popups), or if it’s worth a new tab on tap/click, etc. It is a hassle to configure, but then it usually just works. I suspect that your browser has to be running anyway, because a service worker is just a restricted browser env. Push providers may have delayed delivery, at least there is a ttl field in a notification at server-side. Didn’t test that much and only used that in one project, but remote alpha users already reported missing notifications. I have no clue how to debug that, because it simply doesn’t happen (phone config seems okay).