Basically you can PULL or PUSH.
WP Rest API – PULL
Use the REST API to query for posts. Look for the latest publish date and verify your last know post. If there is a difference, send a message or trigger a notification in the app. It’s possible you could set the last known on the user’s metadata and use that to dif.
add action('publish_post'); – PUSH
Hook the publish post then use PHP to trigger an SMS service. You could at this point query all users to get all known information for SMS messages.