Did it ever become "public" knowledge that Anakin was Darth Vader, after he died? There were of course several people at the
Viking Jedi funeral of Darth Vader, but did the others (not including Leia) know he was Anakin and the father of Luke and Leia? EU or canon?
Luke told Leia. Leia told Han. Leia and Han told Ben Solo. That may be it.1 Leia isn't even sure if Chewbacca knows. So speaketh the new canon novel, Bloodline.
Note: Spoilers for Star Wars: Bloodline ahead
Casterfo frowned. “Princess Leia?”
He had dared to tell her his most painful truth. She could never reveal hers, not to anyone who didn’t already know; Leia understood that. But perhaps she could find the courage to match his honesty with a measure of her own.
“At the beginning of the war against the Empire, just as the Imperial Senate was dissolved—” She swallowed hard. “My ship was captured by the Devastator. That was Darth Vader’s flagship at the time. He personally brought me to the Death Star, where he— where he questioned me.”...
“Then he brought me to witness Alderaan’s destruction. Vader’s hand gripped my shoulder just after I watched my planet die. He made me suffer in every way a human being can suffer, all for the love of the Emperor.”
Casterfo slipped his arm from under her hand — she had gone utterly motionless — and grasped her fingers in his. As old as she was, as cynical as she’d become, Leia would never have guessed that such a gesture could still move her, but it did.
“I hated him so much,” she whispered. The breeze blew past them, rustling the blueblossom trees within the hanging gardens. It was as if they were helping to hide her painful words. “Sometimes I felt as if the only thing that kept me going in the aftermath of Alderaan was the strength of my hatred for Vader.”
For my father.
- Star Wars: Bloodline
Think of your conversation with Casterfo as practice, she told herself. One day she would have to reveal all this to her son. The truth of Vader’s identity had shattered her; she could not imagine what it might mean to Ben. At least Luke could tell Ben the most important part— that Vader had, in the end, been redeemed. Anakin Skywalker had returned; the dark side had been defeated by the light.
Tai-Lin hesitated before he came closer. “There’s something I’ve been meaning to ask you for a long time, and finally I feel I know you well enough to dare. If I overstep my bounds, please, tell me.”
Uh-oh, Leia thought. Was this some kind of romantic overture? Surely not. She seemed to have a case of paranoia this morning. But she understood why an outside observer might believe she was available, which was proof Han had definitely been gone too long. “I’ll tell you, Tai-Lin. Go ahead and ask me.”
“Did you never consider following in your brother’s path and becoming a Jedi?”
Leia found herself caught short. “Why do you ask?”
“They say on my world that the Force sometimes runs strong in certain families.”
So much of the lore of the Jedi had been lost— but on Gatalenta, the old religion had remained strong. History had become legend, but some of the legends were still told. Gatalenta had been one of Luke’s first destinations when he began his research into the Jedi Knights of old.
Tai-Lin continued, “If that is true, then you might have the potential, just like your brother.”
And my son, she thought but did not say.
“If you have that ability, then I cannot imagine why you would not become a Jedi as well,” Tai-Lin finished. “Surely I’ve known few people who would make a finer Jedi Knight than you.”
Leia inclined her head in gratitude for the compliment, but she could not answer right away, because she could not tell the full truth. The Force was too important a subject to be shared lightly, even with Tai-Lin, her ally and friend.
However, it wouldn't have been necessary for Luke to tell anyone else - the secret had been exposed in front of the galactic senate by a political rival of Leia's:
Ransolm continued, “Princess Leia’s lies have protected her long enough. Her deception cannot be permitted to endanger the entire galaxy. If people are considering electing her as First Senator, they have the right to know exactly who they’re voting for.”
Leia’s confusion crystallized in an instant around one of her most terrible fears: He knows.
No. Impossible. Nobody had ever known this besides her, Luke, and Han; she wasn’t even sure whether Han had told Chewie. They hadn’t even told Ben yet. So Ransolm couldn’t have learned the most horrible truth of her life. There was no way. He had to be talking about something else.
But what? There was nothing Leia could think of, nothing besides...
Ransolm pointed at her and declared to the entire Galactic Senate, “Senator Leia Organa is none other than the daughter of Darth Vader himself!”
How did he find out, if Leia, Luke, and Han didn't tell him? Oddly, he had obtained a recording of Bail Organa spilling the beans - a recording Leia had never heard.
Uproar swirled around Leia, surrounded her. She could hardly hear the shouts, stamps, whistles, and pounded desks from the senators over the rushing of blood in her own ears. Her breaths came shallow in her chest, as if Ransolm Casterfo’s revelation had wound itself so tightly around her that she would soon suffocate.
“This is a lie!” Varish howled over the din. “A filthy, outrageous lie, and one Senator Organa will rise to deny!”
Will I? Leia’s thoughts filtered through a daze. Her limbs had gone so watery and weak from shock that she wasn’t sure she could get to her feet.
“I do not come without proof,” Ransolm said. “I will now present my evidence for everyone to hear, so that they can all know how close we came to allowing Lord Vader’s daughter to rule over us all.”
Proof? What possible proof could there be? Leia stared at Ransolm, aware that she should feel angry or betrayed, but unable to summon any emotions besides horror and confusion.
Then Ransolm held up a box — not just any box, a keepsake chest. Every child on Alderaan had one. Parents and grandparents carved the designs, but only the child decided what would be put inside. Placing one of your possessions inside the keepsake chest meant that you had outgrown it but recognized its importance to you. In adulthood you could open the chest, look back, and see how you had chosen to tell your own story.
Leia thought that keepsake chest looked like hers. But she hadn’t seen it in at least thirty years, and surely it had been destroyed with Alderaan.
Nobody was ever supposed to open a keepsake chest without permission, but Ransolm opened this one now. From within he pulled out a music box, one Leia recognized so instantly that the memory pierced her heart like an arrow. She had no time to wonder how it had survived or come into Ransolm Casterfo’s possession; Ransolm had already opened it, and the tune began to play. Mirrorbright shines the moon—
— and then Bail Organa began to speak. Just the sound of his voice brought tears to her eyes, but every word revealed her deepest secret. In despair Leia thought, They are using both my fathers against me.
Bail Organa, who had so often spoken out in the Old Republic and Imperial senates, who had possessed the courage to stand against Palpatine when nearly every other planetary leader had bowed to the Emperor’s power, uttered his last words to the public from the music box, played on every speaker, to be reproduced by every news source around the galaxy within moments. “Your father has become Darth Vader.”
The shouting rose again, even louder than before. Leia bit the inside of her cheek, struggling to keep some small measure of her composure. Her father — her real father — had had the foresight to store this somewhere offplanet. He had given Leia the truth in the only way he could. And the Senate had repaid his service and his love by using it to humiliate his daughter. She felt a moment of dull gratitude that at least Bail had never known this; he’d never had to face just how terribly his message had been used against her.
1 Of course, R2-D2 knew by the end of Revenge of the Sith, and unlike C-3PO, his memory wasn't wiped. Even if it had been, R2 was on Dagobah when Yoda confirmed to Luke that Vader was Luke's father, so he may have overheard it then. So we may be able to add R2 to the list.
Luke tells Leia in ROTJ, right before he also tells her she is his sister.
My father has it, I have it, my sister has it. Yes, it's you, Leia!
Edit: You now clarified that you mean "besides Leia". I'm not aware of canon events where Luke would have disclosed that.
From TFA, we do know that Han knew
There's nothing we could have done. There's too much Vader in him (1:24:26)
As did Lor San Tekka, almost certainly, as he know Ben/Kylo's heritage
The First Order rose from the dark side, you did not. (0:7:18)
Yes, at least some people are aware of this. All the Solo/Skywalker family is aware of this, as well as the (new) Jedi order. I have no precise sources but it should be referenced in many books.
One of the first reference could be in the Thrawn saga where Noghris called Leia "Lady Vader" but I think it was more or less secret, then just became known by more people.
In Disney canon
In The Force Awakens, yes
Kylo Ren is aware he's Darth Vader's grandson.
There is a scene where he speaks to Vader's helmet :
Show me again the power of the darkness, and I'll let nothing stand in our way. Show me, grandfather, and I will finish what you started.
Regarding other people, there is no exact answer yet as far as I know, but some details could be provided in the upcoming books, comics and movies (note: I heard rumors that we may have some info in the Bloodline novel but this is not confirmed as it hasn't been released yet).
In the first Black Fleet Crisis novel, it seems to be common knowledge, at least among the political/military decision council that President of the Senate Leia Organa Solo is Darth Vader’s daughter. Most people in the room (including Ackbar and Wedge) don’t seem to think it’s a big deal. This is one of the Bantam publisher novels from the 1990s. So, Legends.
It doesn't seem to be common knowledge and would have been very easy for Palpatine to keep the details of the massacre of younglings concealed (Jedi/present company such as yoda and so on excluded) but if anyone has watched rebels you see that through visions Ashoka finds out about Anakins wrong doings but who she passes that knowledge onto I can't tell you but the fact that Capt. Rex never says anything and the others don't even know who Anakin is its safe to say other people didn't know
Edit I jus realised this was an answer to a comment from T.E.D but still Im not sure if Luke told others but as some else said Han Solo clearly knew and therefore chewy too.. I can't see much reason why they would need to keep it a secret though the empire has fallen vader is gone and the emperor is no longer hunting the remaining Jedi as to the first order it doesn't seem that destroying the Jedi is their primary goal even if Kylo Ren does