The truth: that he was living with Muggles.
We only get brief snippets of what people thought he might have been up to, when he’s just on the cusp of re-entering the magical world. Once he’s at Hogwarts, the shock factor of his mere presence wears off a little.
There are a few things that were definitely well-known before he came to Hogwarts:
He was living with Muggles
There are several references to this:
It’s not clear whether people realised that he’d gone to live with Muggle relatives, or merely some Muggles that were trusted to look after him.
He had a lightning-bolt shaped scar on his forehead
We also know that people were aware of the lightning scar as a distinguishing feature, before he rejoined the magical world. When he shows up at Hogwarts, everybody wants to see it, and the Weasleys seem to be well aware of it:
As Fred and George help him get his trunk onto the train:
With the twins’ help, Harry’s trunk was at last tucked away in a corner of the compartment.
‘Thanks,’ said Harry, pushing his sweaty hair out of his eyes. “What’s that?” said one of the twins suddenly, pointing at Harry’s lightning scar.
“Blimey,’ said the other twin. ‘Are you –?”
“He is,” said the first twin. “Aren’t you?” he added to Harry.
— Philosopher’s Stone, chapter 6 (The Journey from Platform Nine and Three-Quarters)
And when they tell their family, both Molly and Ginny seem to be aware of it:
“Oh, Mum, can I go on the train and see him, Mum, oh please ...”
“You’ve already seen him, Ginny, and the poor boy isn’t something you goggle at in a zoo. Is he really, Fred? How do you know?”
“Asked him. Saw his scar. It’s really there – like lightning.”
— Philosopher’s Stone, chapter 6 (The Journey from Platform Nine and Three-Quarters)
He’s definitely alive
He gets spotted in the street (either by resemblance to James, or from his lightning scar). Not only would this confirm the story that he was living with Muggles, but it would also be a strong suggestion that he was still alive.
When Harry first visits the Leaky Cauldron, he meets one of his admirers:
“I’ve seen you before!” said Harry, as Dedalus Diggle’s top hat fell off in his excitement. “You bowed to me once in a shop.”
— Philosopher’s Stone, chapter 5 (Diagon Alley)
Given how excited people seem to be to see or hear of Harry, I’m sure a story about meeting him on the street would have gone down well in the pub. I imagine Dedalus Diggle wasn’t buying his own drinks for quite a while.
And earlier in the book, we hear about a whole laundry list of such people:
Yet sometimes he thought (or maybe hoped) that strangers in the street seemed to know him. Very strange strangers they were, too. A tiny man in a violet top hat had bowed to him once while out shopping with Aunt Petunia and Dudley. […] A wild-looking old woman dressed all in green had waved merrily at him once on a bus. A bald man in a very long purple coat had actually shaken his hand in the street the other day and then walked away without a word. The weirdest thing about all these people was the way they seemed to vanish the second Harry tried to get a closer look.
— Philosopher’s Stone, chapter 2 (The Vanishing Glass)
