I'm not sure what the difference is.
If you are talking about a push notification that appears in the notification bar on the android device there is no separation between "message" and "notification" they are the same thing.
There are 2 types of notifications on Android devices:
push notifications - you download an app, the app registers with the OS that it will receive push notifications, a server sends out notifications to the Google Cloud Messaging service, that service sends the notification to the devices over a wifi or mobile network.
local notifications - the app runs in the background and creates a notification when it receives certain information from a network or a timed event such as "it has been 24 hours since you played your health is restored". This is unavailable via AppGameKit for Android.
Both types require network connectivity. How else would one player communicate to another?
Facebook Messenger uses a hybrid, I believe. It receives a push notification with special data that wakes the app, the app then turns that data into a local notification with pretty overlays and such.
Most applications use only push notifications. It is composed of a notification that appears in the notifications bar and a message. Then when you tap the notification it will launch the app. The app will then look at the notification data and interpret it, usually it is a signal to obtain message text that was sent. Unfortunately with AppGameKit, we can just do a push message and the app cannot do anything with the data.
In my game, Wordspionage, we use push notifications to tell players that they've been invited to games, it is their turn, and other events. When they tap the notification it opens the app. When the app opens the app has to connect to our server to download the correct information.
If this is not what you are talking about please be more specific and I'll do what I can to explain further.