Your native app must be live in order to send push notifications. There’s no way for users to receive them without having the live app installed on their device.
For Web Apps your users must accept push notifications when they visit your PWA page. They do not need to have the app installed nor do they need to have it or the browser running.
When an app is installed, users can choose to enable or opt out of push notifications. If they opt out, they won’t receive any from your app.
If you want to receive push notifications successfully, it’s important to enable both push notifications AND location services, which allows you to receive location-based messages.