Hi,
Acymailing doesn't send a notification when the mailing has been send. When you go to Newsletters you alsways can see the percentage that has been send. If it's less then 100% the sending is still running. Alternativly you can have a look at your queue.
That leaves you with identifying sending errors and bounces. You'll find the sending errors in your cron log. Acymailing handles the bounces for you. You can edit each bounce rule to notify you or to subscribe your user to a special list and have a look at the users in that particular list. Alternativly you'll find the bounces also in your statistics. You would however need the Enterprise version to be able to use these options.