As of WordPress 4.6 it is possible to hook into the wp_send_new_user_notifications action to disable the new user notifications send to site admins or the new user.

// The Parameter "$behaviour" can be set to:
// "none" (no notifications are send); 
// "default" (no changes); 
// "admin" (notifications send to admin only); 
// "user" (notification send to user only); 
// "both" (notifications send to admin + user)
public static function limitNewRegistrationNotifications($behaviour){
    // do nothing
    if ($behaviour == 'default'){
        return;
    }

    // handle user registrations (self registered users)
    remove_action('register_new_user', 'wp_send_new_user_notifications');

    // new users added via wp-admin are created using add_user() -> edit_user() chain, NOT register_new_user()
    // @see https://developer.wordpress.org/reference/functions/add_user/
    remove_action('edit_user_created_user', 'wp_send_new_user_notifications', 10, 2);

    // notifications disabled ?
    if ($behaviour == 'none'){
        return;
    }

    // add custom callback and override the $notify setting with custom behaviour
    add_action('register_new_user', function($user_id) use ($behaviour){
        // trigger notification
        wp_new_user_notification($user_id, null, $behaviour);
    });
    add_action('edit_user_created_user', function($user_id) use ($behaviour){
        // trigger notification
        wp_new_user_notification($user_id, null, $behaviour);
    });
}

 

This Tweak is available as part of the Tweakr WordPress Plugin.