Some people did. But not those who knew Dumbledore.
As NominSim says, there are differing opinions about this in the wizarding community. Some believed he was dead, some believed he was planning his return to power and some believed he was gone forever (even if he wasn't dead in the technical sense of the term). One person who did know (or at least guessed with unerring precision) what happened to Voldemort was Dumbledore.
"Did I believe that Voldemort was gone forever? No. I knew not whether it would be ten, twenty or fifty years before he returned, but I was sure he would do so..."
(Order of the Pheonix, Chapter 37, The Lost Prophecy)
Dumbledore was convinced that Voldemort would come back at some stage. Here he's recounting to Harry his thought process when he decided to send Harry to the Dursleys. So, even at this very early stage, Dumbledore recognised the significance of Harry's scar and knew that Voldemort wasn't truly defeated. As b_jonas points out, Dumbledore may very well have made his suspicions public. Hagrid, as someone who looked up to Dumbledore, would most likely have taken his word as gospel.
There's also the mysterious events surrounding Voldemort's demise to consider. Many witches and wizards seemingly knew the rough outline of what happened, as is shown by McGonagall knowing the rumours even having spent the entire day perched on a wall, disguised as a cat. This answer also shows that people knew some basic facts about Harry. People knew that Lily and James were dead, that Voldemort was gone but that Harry survived. That's an enigmatic cocktail of strange circumstances. In the absence of firm evidence, even those who didn't know Dumbledore may have speculated that Voldemort was only temporarily thwarted - how could a baby have finished him off for good?
@AncientSwordRage makes a good point in this answer that the Ministry believed Voldemort to be alive.
"The four of you stand accused of capturing an Auror - Frank Longbottom - and subjecting him to the Cruciatus Curse, believing him to have knowledge of the present whereabouts of your exiled master, He-Who-Must-Not-Be-Named -"
(Goblet of Fire, Chapter 30, The Pensieve)
Additionally, Voldemort reveals that the Ministry was actively looking for him.
"But I dared not go where other humans were plentiful, for I knew that the Aurors were still abroad and searching for me."
(Goblet of Fire, Chapter 33, The Death Eaters)
Finally, at least a few of Voldemort's followers believed whole-heartedly that he would return.
"The Dark Lord will rise again, Crouch! Throw us into Azkaban, we will wait! He will rise again and come for us, he will reward us beyond any of his other supporters! We alone were faithful! We alone tried to find him!"
(Goblet of Fire, Chapter 30, The Pensieve)
However, you only have to look at the reaction of the public during Order of the Phoenix to see how many people believed that Voldemort was "gone for good".
"While the Ministry insists there is nothing to fear from Voldemort it's hard to convince people he's back, especially as they really don't want to believe it in the first place."
(Order of the Phoenix, Chapter 5, The Order of the Phoenix)
People were basically in denial because the thought of Voldemort's return was so terrifying to them. And as long at the government and the media had the same story they were happy to believe he was still dead, despite evidence to the contrary.
So why were the witches and wizards so happy in Philosopher's Stone?
It may be splitting hairs but this is the difference between Voldemort being "gone" (as McGonagall puts it) and "gone forever" (as Dumbledore later puts it). For your average witch or wizard who's been cowering under the threat of death, torture and oppression, it didn't matter. The danger had passed, and that was all they cared about. Hence the owls, fireworks, street parties etc. And they were right to - for 13 years it didn't make any difference whether Voldemort was dead or not. He was gone and that was why they partied.