Our organization has numerous teams and a lot of repositories. It appears there are only two ways to prevent a user from getting notifications (specifically on pull requests):
Revoke their access to the repositories that they don't want notifications for -- this really doesn't work because we want them to have access to the code, but they don't need bombarded with pull request notifications.
Turn off all notifications at the user settings level -- this really doesn't work because then they won't even get notifications for projects that they are actually working on.
Additionally, all of these notifications fill up the inbox very quickly and render it pretty much useless. You have to go through a page at a time and delete the messages to get rid of them. There should be an option to delete all messages, not just the ones on that page. (More control over what you get notified on would also help mitigate this)