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Originally, the notification settings were working. However, after I updated from 220.127.116.11 (yikes) to 2.1.11, this feature stopped working. I found this thread: http://vanillaforums.org/discussion/25706/solved-what-in-the-db-could-prevent-email-notifications-of-pms-being-sent which helped me trace the problem to the GDN_Permission table.
The GDN_Permission table indeed appears to be the problem, because when I import a new 2.1.11 GDN_Permission table, the feature works again. The problem is I am using the old 18.104.22.168 vf_Permission table (which I renamed to GDN_Permission, along with changing all of the other vf_ prefixes to GDN_ prefixes) to get 2.1.11 working.
I was able to further narrow down the problem to the Garden.Email.View column of the GDN_Permission table, thanks to @Linc 's and @peregrine 's comments in the above discussion. When I manually change a 0 to a 1, then the user starts getting notification emails sent successfully again.
If someone could help me figure out how to re-connect my database to the notification settings in Garden.Email.View that apparently are not getting updated correctly, I'd be grateful. I'd also be interested in learning how I can determine which user refers to which row in the GDN_Permission table so I can manually update their notification preferences if necessary. I see there is a PermissionID unique key, but I am not sure how to determine which user each PermissionID refers to.