Push notifications in a local network
First if you decide with the XMPP protocol .. you can use any of the servers you mentioned and use any library for the mobile Smack for example for android, the mobile library will automatically update you with events using listeners .. it's a software layer that handles notifications.
Second if you want to implement it manually .. you'll have a service running on the mobile device that checks for updates periodically by requesting them from server.
Third The actual push notifications need to be registered over the internet, android for example need to communicate with Google Cloud Messaging.