Very few people knew about the true identity of the cyborg known as the Darth Vader. It is obvious that Obi-Wan, Yoda, Palpatine and Darth Vader himself knew what became of the 'Hero with No Fear' after Order 66.
However, it is also clear that Owen and Beru Lars knew of the demise of Anakin Skywalker and his rebirth as the Emperor's right hand man. It is not confirmed for certain as to in what detail they knew of his fate, but many believe that during Obi-Wan's exile on Tatooine, he may have revealed some details to them. But how much he disclosed to the Lars is unknown; as we see in A New Hope, Owen clearly has a sense of disgruntlement towards the hermit Ben Kenobi, who seeks shelter on the planet.
Furthermore, it is also quite possible that Bail Organa knew about Darth Vader's identity before Order 66. As he was there for the birth of the twins and witnessed Padmé's death, supposedly because she 'lost the will to live'. It is sensible to assume that because Bail did know Padmé quite well, he could have come to the conclusion that the father was Anakin, if Obi-Wan or Yoda hadn't told him already, and that the cause of her heartbreak was something to do with him. Finally, the timescale also matched; as Anakin Skywalker died, Darth Vader was born, not to mention his ability to wield a lightsaber, which was unmatched by anyone.
But, in conclusion I don't think that many people were able to piece together the shattered parts in order to determine who Vader was before the birth of the Empire. It is also possible that information on such people could have been deleted from record so as to ensure certain articles of information remained a secret.