the Push notification (that will be in core for 2.7, in 2.6 is a contributed plugin) system is connected to the messages/notification system in Moodle.
A user can configure his messaging preferences in Settings -> Messaging, there he can select which type of notifications wants to receive and how (email, jabber, popup, jabber)
So, if you send a private message, the user will only receive it if:
- You have the mobile notifications plugin enabled and configured
- He have the app installed
- He have configured his preferences in Moodle for receiving notifications