In my email app, emails I read are automatically marked as "read" after a few seconds. I can manually press the "archive" button once I've dealt with the email, moving the message from the inbox to the archive and hiding them from view.
This fits my workflow quite well, and I have been wishing for something like this in Trello for quite some time.
I've found that if I have multiple unread notifications, it's very easy to lose track of some of the earlier notifications:
Clicking on the bell icon in the upper-right hand corner of the screen seems to dismiss all the notifications, which makes it hard to keep track of which ones I've already dealt with.
Going to my notifications page let's me see all my notifications, but there's no way to tell which ones are considered read or not, and there's definitely no way to see which ones I've dealt with or not, since I can mark a notification as "dealt with" (the equivalent of archive in an email app).
Enabling email notifications almost fixes this problem, as I can then use my email app's archive feature, but it looks like Trello does not email you notifications that you happen to read on the web version.
