How do you test push notifications on native applications?

I would add:

  • The user is logged in on multiple devices
    • in the same state of the device and app
    • in the different state of the device and app
    • different type of devices (hardware, OS )
  • The user receives before/after/at the same time notifications from other apps

Also I would try to figure out how much of this is OS related and how much depends on the app.
Do not test too much what is basic OS stuff.