How can I implement a notification system in wordpress?

If you look through the definition of wp_insert_post(), you’ll see

do_action( 'wp_insert_post', $post_ID, $post, $update );

at the end of it. So you can hook into this and perform whatever task you want to after creating a post.

Something like this in your theme’s functions.php or in your plugin file:

do_action( 'wp_insert_post', 'send_notification');
function send_notification($post_ID, $post, $update){
    //your code here
}