I did some googling on this this morning and it appears that other apps are having problems too. Gmail app users seem to have had the problem appear before we had it appear here.
Somebody on one of the threads I was reading this morning said it appears to be a conflict in where a notice is viewed. For example, you get a notification on your phone and then you read the message on your computer - even though the forum has updated the private message as read, the app still somehow thinks it's not read yet.
On that thread, some people had success by going in on the web platform, marking the affected messages as unread, and then going back in to their iphone app and viewing the messages on their phone.
I've just tried this with all of the PM's I was getting repeat notifications for. I'll update the thread later tonight once I know if it has permanently stopped the notifications or not. It's kind of a pain to fix, but if it did fix it, then it lets me have the app on my phone again.
eta: Disabling notifications doesn't stop them, and even logging out of the app doesn't stop the notifications. I had to delete the app to stop the notifications, and within minutes of it being reinstalled the first batch of notifications came through. I hadn't even logged in at that point...
eta again: The repeat notifications started again a couple hours ago. Looks like that trick isn't much of a fix after all.