That’s a good question. Here’s the thing, there’s people who know parts of the story – Howland Reed told his kids about the Knight of the Laughing Tree, so they may well have guessed that Lyanna ran away with Rhaegar, but I don’t think he told them about the Tower of Joy (it’s possible Jojen saw it in a vision) although they may have put two and two together

Likewise, I’ve always been under the impression that the older folks at House Dayne know, since Ned stayed there with baby Jon until Jon was able to travel and their servant Wylla was Jon’s wetnurse and likely Lyanna’s midwife – after all, Arthur Dayne was the closest to home of any of the knights at the Tower of Joy, and he’s definitely going to send to home for a servant to help with the birthing of a royal child since that’s completely outside his wheelhouse. Also, this always made sense to me as both the simplest and most tragic answer to how Ned knew where to find Lyanna: Ashara Dayne told her lover where to find his sister, and that’s a big part of the reason she took her life. 

As for Rhaegar, he may may have told some of his closest confidants like Richard Lonmouth, but I doubt it. And I have no idea how he was planning that particular revelation.