"Had the same problem and got mines to work. Here's the problem. Go on the child's device and go to Settings -> iTunes and App Store (NOT iCloud), and see if the CHILD'S ACCOUNT is logged in, or yours (or any other adult). If its not the child's account, there's your problem. Log out whichever adult is logged in on the child's device, and log the child's iCloud account in."
This was my issue. My daughter was logged in under her own iCloud account, but her App Store was still on our account.
A few things about this seem like a bug:
1. If it's not set up correctly for a child, why does it bother saying it needs to ask for permission. It seems that if she's using the parent App Store account, she shouldn't need permission. If permission is based on her iCloud account, then why does it bother checking the app store account?
2. It did send notifications, but only to my wife's account. She was set as "Parent", but me as the organizer, did not get notifications.
Logging my daughter out of my app store account and logging in with her own account did fix the issue, though.
And as has been pointed out in other posts here, if the notification goes by too quick, you should still be able to find it by pulling down from the top of your screen and swapping from "Today" to "Notifications".