The following tutorial will demonstrate the details of publisher notifications. *Please click on images to enlarge them*
First click Notifications in the menu bar on the left. This shows you a list of your notifications. The notifications only have two columns: Message and Created at (time notification was sent). This is where the publisher can check notifications they have received.
To manage which notifications are sent and to where is in another section. To get there simply click My Account then Notifications.
Here is a list of all the notifications that can be set. They are broken into two sections, Offer and Actions. As you can see, the In-App notifications are all set to fire whereas all the Email notifications are turned off.
You can switch where the notifications go by turning on Email notification and turning off In-App ones. You can also keep them both on, or turn both the email and In-App notifications off.