Drupal 7: notify user module recommendation
For the record: The Notifications module will not do what you want.  But worse, the D7 port is in alpha and hasn't progressed since 2011-Dec-04, there are 217 bugs in the issue queue and no maintainer in sight. Its maintenance status is listed as "Seeking new maintainer".  In other words, this yet another abandoned Drupal module. Unless you're prepared to adapt and maintain this yourself - stay away.

There is a survey page for notifications modules, but it only confirms that what is called a "Notification" module by Drupal developers provides email notification, which is not what you want.  The type of notofication you want tend to be referred to as "flags" by Drupal developers, so you want to search for "flags" in addition to "notifications".

To get on-screen Javascript notifications (similar to Facebook and Stack Exchange), my suggestion for a starting point is to see whether you can combine the Flag module, the Message module and the Rules module to do what you want.  But to get these three modules combined to produce the desired effect will not be trivial.

An alternative solution (first mentioned in a comment by Nigel Waters), you could use the Message module to create the activity stream. Then use Views to generate the stream applicable to the currenly logged in user. And finally use the Menu Badges module to display the number of notifications.

You may also want to read: How to create a basic message - notifications system displaying a real time message counter badge?