No; Henry (and the rest of the OUAT cast) knows that the people in the sleeping curse are in that nightmare, but until reports start trickling in, no one else will know :( Also, it's worth mentioning that the nightmares can (optionally) continue even after the event.

As far as knowing what's happened to them, nope they won't know what it is until they wake up and the person who did the waking explains. Fun explanation times all around!