Please have a look at react-native-push-notification#silent
Android:
If your Android app is not running when a silent notification is received then this library will start it. It will be started in the background, however, and if the OS starts your app in this way it will not start the react-native lifecycle. This means that if your notification delivery code relies on the react-native lifecycle then it will not get invoked in this situation. You need to structure your app in such a way that push notification. configure gets called as a side effect of merely importing the root index.android.js file.
iOS:
The crucial bit of an iOS silent notification is the presence of the "content-available": 1 field.