Being a Star Wars fan, I give the various issues brought up in the movies a lot more thought than most people. Why? Because I find them interesting. One such 'issue' is the subject of Darth Vader's true identity and just who knew about it- both during and after the movies.
It seems to me that during the time of the movies, only a few people knew who Darth Vader truly was:
The Emperor knew, of course, because he was there during Vader's 'birth'.
Obi Wan and Yoda obviously figured out who Vader was. But I don't think either of them told anyone. I suspect that Kenobi did tell Luke's aunt and uncle, however. The whole "He's too much like his father/That's what I'm afraid of" dialogue could very well have been in reference to this. It would also seem like a prudent thing to do if Kenobi wanted to impress upon them how important it was to keep a low profile. But then again, if he did know, why didn't he change Luke's last name... This all plays back into my thoughts about Ben planning to tell Luke about his heritage at a very specific time (perhaps even AFTER he had killed Vader).
Bail Organa may have suspected the truth about Vader, but it doesn't seem that he had any contact with people who knew the truth. He adopted Leia and parted from Yoda and Kenobi before 'Vader' became publicly known. Based on the events of Episode IV, it seems that Organa and Kenobi had no contact during all the years that Leia was growing up. It is possible that Bail intended to tell Leia of her true heritage at some point, but if he did, then he put it off too long. Besides, how do you tell someone you love that their REAL father might be a monster.
Artoo Deetoo may have had some idea of the true identity of Vader. He was pretty clever and, unlike Threepio, never had his memory wiped. Even so, he never had any direct contact with Vader and so had no chance to prove or disprove any suspicions he may have had.
Outside of these people, it seems that nobody else in the Galaxy knew Vader's true identity. He must have just suddenly 'appeared' as a servant to the Emperor, slowly gaining recognition in his new identity.
This changes when Vader reveals his relationship to Luke in the Empire Strikes Back. But it seems that Luke tells no-one of this dark secret. At first he may have done so out of disbelief- he thought Vader was trying to trick him. Later, however, he seemed to have done so out of caution and perhaps even fear. When people find out your father is a monster responsible for the deaths of millions across the galaxy they're liable to have some strong reactions. Indeed, when Luke finally does tell Leia in Return of the Jedi, you can see the horror on her face- and the pity she feels towards Luke. The horror only deepens when Luke reveals that Leia too is the child of Anakin/Vader.
Where things really get interesting, however, is AFTER the movies. In the 'expanded universe', the whole issue seems to have been glossed over or ignored. The only mention I recall of it is when the Noghri somehow recognize Leia's relationship to Vader via her scent. Other than that, it just doesn't seem to come up much. This I find very odd, especially with Luke and Leia taking such prominent positions in the New Republic. In my mind, there are only two options for what is going on here:
Option A: Luke and Leia have kept their relationship to Vader a secret. Nobody outside their close circle of friends knows the truth. Thus, most people just see them as the descendents of the Jedi Hero Anakin Skywalker, who died tragically (but obscurely) in the turmoil of the end of the Clone Wars.
Option B: Luke and Leia have revealed their secret to the Galaxy at large- that Vader was the corrupted (and then redeemed) Anakin Skywalker and that they are both his children.
Both options seem problematic to me. In option A, both Luke and Leia are harboring a 'dark secret'. This would have to be very stressful for both of them. For Luke, as a Jedi master, this could be damaging of his ability to teach or indeed continue his own learning. In my opinion, being a Jedi is about being honest and open- about confronting problems rather than avoiding them. By keeping this secret, Luke would always have to hold some part of himself back- and would always have a nagging fear of the truth coming out. For both Luke and Leia, this secret would be a political time bomb just waiting to go off. If they hid the fact they were Vader's children and it came out some other way it would look like a conspiracy and could cause the public to doubt them, despite their obvious heroics during the rebellion.
If Option B is true, then the Galaxy (as depicted in various novels) has been incredibly understanding of Luke and Leia’s heritage- to the point where even political enemies don’t seem to use it against them. I find this a little difficult to believe. A guy like Borsk Fey’lya, for instance, seems to be the type who’d exploit that particular ‘weakness’. And yet he doesn’t do anything of the sort- and Leia actually becomes Chancellor of the New Republic without the whole issue of her father ever coming up.
All that having been said, however, I tend to lean towards Option B as being the case. To me, it seems to be the most logical. As far as that being a political weakness is concerned, Leia has a huge trump card to play: The fact that her adopted homeworld was destroyed by the Empire. This, in addition to her well-documented role in the victory of the Rebel Alliance would make her rather bullet-proof when it came to attacks regarding her parentage. She may be a blood relative of Vader, but she has proven where her loyalties lie. The same can apply to Luke- considering his own role in the downfall of Vader and the Emperor. There is also the fact that the Rebellion WAS a civil war. I doubt if Luke and Leia are alone in having parents or other relatives who were on the ‘other side’. Yes, their particular parent may have done much more harm than the typical Imperial Officer, but overall, their situation was probably quite common.
So, in my Star Wars universe at least the ‘cat is out of the bag’ in regards to the identity of Anakin Skywalker as Vader. While he is probably reviled by many (and rightly so), his story was not hidden from the public. It serves as both a cautionary tale and perhaps a source of hope that even the most evil might find enlightenment and do what is right when it counts.