Harry Potter is well known to be the only one to survive the Killing Curse. He is famous for being the Boy who Lived, his scar is well known.
So my question is, how did it become known that he defeated Voldemort? How did it become known that he has a scar?
We have four people present that night in Godric's Hollow. James and Lily are dead. Voldemort is mostly dead, we are not sure whether at the time he knows what happened, but he is not available for questions anyway. Harry survived, but is too young to understand what happened, and too young to talk, besides nobody asks him. So why would anybody think that Voldemort cast a Killing Curse at Harry, hit him, and Harry survived that curse? How would anybody know about his scar? How would anybody know that the scar is the result of a failed Killing Curse?
As Bellatrix below points out, it’s possible that Dumbledore could have helped spread the rumors himself. That is a reasonable assumption, because Dumbledore is usually well informed. He knew that the Potter parents were dead and that Harry lived before he sent Hagrid there, and Hagrid was there before Sirius. If it was Dumbledore, what would be his motives be?
Note that Hagrid brought Harry directly from Godric's Hollow, so the scene about the scar that Bellatrix quoted is the first time Dumbledore saw the scar.
Radhil mentioned in a comment that Hagrid was not known to keep secrets. As Dumbledore knows Hagrid and Dumbledore sent Hagrid to fetch Harry, that means that Dumbledore knew that would happen and it was either his intention, or at least he didn't want to avoid it.
But the part of Hagrid spreading the news would only explain the information about the scar. Hagrid has no way to know what happened before he arrived.
@TheWasp I see you changed the title, I added "about Harry's part" because that is the main focus of the question.