Own push notification gateway

Ok, I understand that the only way out is to compile the application yourself.

I would like to ask if the lack of push completely cuts off the possibility of notifications in the application? I understand push notifications as an overriding notification that should appear on the phone immediately. The notifications themselves, however, operating from the application level (in the background) should still communicate about the incoming message. Or so?

To what extent can the lack of push make our life difficult?