That’s not how it works. Not on iOS anyway.

For a standard notification the content of that notification is sent through the push notification servers. This includes the title, text, icon, grouping, and sound presets to use. The majority of user-visible notifications are sent this way - the app on the device does not run.

That allows the OS to display your notification without ever running the app, which saves limited resources on the phone. Originally this was the only option, a push notification couldn’t start your app.

These days an app can also register a notification extension which is a standalone program that can modify the incoming notification. It has 30 seconds to do whatever it needs to, though you need to be careful with RAM use or the OS will kill the process and present the notification unaltered. Generally you’d put something generic in the push as a fallback.

There’s also background notifications. These let the app run for 30 seconds and the app can post a local notification during this time, but they’re not guaranteed to be delivered. The OS can decide the system doesn’t have the resources and defer or drop them, or terminate the app before it’s finished if the ram is needed elsewhere.

There are some other special cases depending on what your app does.