We use pre-moderation, so blog authors aren't notified of new comments until after they've approved them. This is of very limited value. Maybe this is simple, and my bloggers are missing it. They want to be notified when there's a new comment, not after they've already approved it. By then, they already know the comment is there.
@Warren_Brill- There are no such settings available in community admin. As there are separate notification for Moderation which you can enable from,
Community Admin - Mod Tools - Send Moderation Manager notifications by email
Have you enabled this for your community?