Answered: Notification does not work properly when sending messages. The message does not actually show a notification.

It's working just fine for me:

Check for plugin conflicts.

As I always say, disable all plugins and only leave the targeting one, in this case OnSite Notifications. Then start activating other plugins one by one. As you activate a new plugin, run a test and check if the messages are still working as they should. Keep re-activating plugins and running tests and eventually you will find the breaking point, in other words, the second plugin that's causing a conflict. Then you can work from there.