AFAIK specific notifications are sent to you (i.e. automatically pushed) when this function is enabled by giving permission.
All threads wherein you make a post will send a notification to your e-mail address whenever someone else posts something in it as well, either as a direct reply to your post or as a general contribution to the discussion. The same goes for threads you haven't posted in but which you have marked as to be following them regularly.
For this function to work properly your e-mail address has to be part of your profile data.