Rubeus "I shouldn't have told you that" Hagrid.

 

Because JK Rowling.

Seriously, news leaked out ridiculously fast. People were celebrating when the only people that knew what went down were Sirius and Hagrid. McGonagall had heard rumours that Harry had taken down the Big V before she even saw Dumbledore that night. Either someone blabbed or it's a plot hole.

 

There are multiple possible reasons. Dumbledore wanted to finish the war - which might have continued for several weeks otherwise. Also, he might've wanted to give Harry a sense of destiny, generated from so many expectations. Another possible reason is that Dumbledore thought the people deserved to know the truth.

EDIT:

I don't think people would've trusted Hagrid's word. He might have said something and Dumbledore confirmed [from the reasons I said or so as to not embarrass Hagrid].

McGonagall meets Dumbledore a full day after, iirc, so It's possible if unlikely that the rumor have reached her.

 

Several wizards and witches - or at least quite a few - are living in Godric's Hollow. Pettigrew removed the Fidelius Charm and the house suddenly reappeared. Then there was some kind of explosion, perhaps even the Dark Mark, so it's likely there are a few eyewitnesses. Someone must have alerted Dumbledore and Sirius, so other people knew about the event. Especially later on when Hagrid retrieved Harry there might have been wizards assembling at the house, and even Muggles started to notice it.

So I think it is likely some people watched Hagrid carrying Harry. There certainly were fast spreading rumours involving Harry, Voldemort and Hagrid, and it is likely the Prophet later interviewed Hagrid.

 

Aurors would likely have to take the statement of Hagrid and Dumbledore. Hagrid for his discovery of Harry - and Dumbledore for his placement with the nearest blood kin of Harry. Dumbledore and certain Aurors likely gave statements to that effect to the Daily Prophet. Honestly, this is an entirely unimportant detail.

 

It's no secret that anything Dumbledore says is taken as damn-near fact in the wizarding world - at that point in his life, at least. It's also no secret that Dumbledore puts a helluva lot of faith in Hagrid (delivering baby Harry, re-introducing him to the wizarding world, the Philosopher's Stone, letting him teach, etc.). Those two aspects combined paint a clear picture:

"Little Harry Potter got a cursed scar? How do you know that?"
"I heard it from [chain of people], who heard it from Hagrid, who was told by Dumbledore himself!"
"Merlin's baggyy-fronts! It MUST be true!"


...or it was that hoe Rita Skeeter. Just sayin'.

 

Maybe JKR is just a shitty author? I mean, McGonagall seemed to know only a few hours after it happened, and where Dumbledore was taking him.

 

I'm thinking it was a concession Dumbledore had to make in order to give Harry the Boy-Who-Lived status. The Headmaster's words might carry a lot of weight but I think people would be a bit skeptical about a baby defeating the Dark Lord who has been terrorizing the country side for years.

Why the belief in a baby and not some trap by the adult Potters that required their heroic sacrifice? Where is Dumbledore's proof?

And since I highly doubt he would have blabbed about the Prophecy, the scar could have been used as that little push to give the victory to Harry and not his parents.