Holo-vids of the destruction of the Death Star and a message from Princess Leia were extensively shared via a hack to the Holo-news network. Those that couldn't see it live would have heard about it or seen a bootleg copy of the announcement.
In short, the majority of the Galaxy's population are aware that the Emperor is dead almost immediately after it happened.
Princess Leia Organa. Once of Alderaan. Now: one of the heroes and leaders of the Rebel Alliance.
The recorded image of the princess speaks:
“This is Leia Organa, last princess of Alderaan, former member of the Galactic Senate, and a leader in the Alliance to Restore the Republic. I have a message for the galaxy. The grip of the Galactic Empire on our galaxy and its citizens is relinquished. The Death Star outside the forest moon of Endor is gone, and with it the Imperial leadership.”
Here the hologram changes to a sight all too familiar to Sinjir:
The Death Star exploding in the sky above Endor.
He knows because he was there. He saw the great flash, the pulse of fire, the bulging clouds like brains knocked out of some fool’s cracked head. All the bits of it up there, still, floating like so much detritus. The image flickers. Then it’s back to Leia.
“The tyrant Palpatine is dead. But the fight isn’t over. The war goes on even as the Empire’s power diminishes. But we are here for you. Know that wherever you are, no matter how far out into the Outer Rim you dwell, the New Republic is coming to help. Already we’ve captured dozens of Imperial capital ships and Destroyers—” Now the image becomes three-dimensional footage of Imperials being led off a ship’s ramp in cuffs. “And in the months since the destruction of the Empire’s dread battle station, we have already liberated countless planets in the name of the Alliance.” A new image: rebels being greeted as saviors and liberators by a cheering crowd of—where is that? Naboo? Could be Naboo. Back to Leia: “Be patient. Be strong. Fight back where you can. The Imperial war machine falls apart one gear, one gun, one stormtrooper at a time. The New Republic is coming. And we want your help to finish the fight.”
Star Wars: Aftermath
On the opposite side of things, the Empire tries to use body-doubles and faked footage to counter persistent rumours of Palpatine's death. Even his own side don't believe it, for the most part.
Jintar kneels. “Bor,” he says. “Listen to me. We’ve been lied to. Adelhard has sealed off the whole sector. Massive blockades with a ragtag Imperial remnant. But that’s not how they keep control. They keep control by lying to us.” He takes a deep breath. “The Emperor is dead, Bor. It’s been confirmed.”
“Lies,” Borgin hisses. “Of course, that’s what his type would have you believe!” He gestures with his chin toward the rebel, Kars. The scruffy pirate in the patchwork armor does nothing but scowl and shake his head. “I’ve seen the holovids. You have, too. Palpatine is alive and well on Coruscant and—”
“He’s just a stand-in. A proxy. An actor.”
“No. More rebel lies.”
“We’ve done the comparison. The vids don’t match. This…person in the dark robes isn’t Palpatine. Different chin, different gestures. A poor facsimile.”
Star Wars: Aftermath