Odoo mobile provides real-time notifications using the Google cloud notification platform. For online users, it is already activated:
When you see the device registration message on the related partner for the login user, it confirms that your device is successfully registered to obtain the real-time push notifications from your instance.
Push notifications are dependent on your devices and on Android; most of the standard devices get the notifications even if the application is not running. On some devices, you will get notifications only if the Odoo mobile application is running in the background. If your phone is rooted, and you have installed the custom Android without the Google Play service, notifications may not work.