Rumors are the single most important tool in a DM's arsenal in a sandbox game. They allow the DM to nudge in one direction or another, without a train pushing the PCs along. They allow the DM to flesh out the world without making the PCs have 'eyes on.' A DM can make a power player known well before the first encounter. The possibilities that rumors give is literally endless.
There is a downside to rumors as well. They can overwhelm players, if there are too many given. They can distract from where the game needs to go. They can make the players over confident. And worst of all, they can become a crutch for both players and DMs.
How do you avoid the pitfalls of rumors while running a game? You need to use them carefully:
1. Make sure not all rumors are true. This helps curb player over confidence. Also, it helps keep rumors from becoming a crutch (at least on the player side). Try to make sure one of the early rumors is false. The earlier the players learn this lesson, the better.
2. Keep rumors vague. A good rumor is something like "I heard the throne is important to explorers of the Citadel." A bad rumor is "The throne in the Citadel has a teleportation rune that brings people back to Var Nae."
The first brings the party's attention to something they might otherwise miss.
The second tells them exactly what to look for, and where to find it. It robs the players of their sense of discovery and accomplishment.
3. Have rumors advance plot, and show the world. Has an army attacked a major city? Chances are there are rumors everywhere about it. Especially in a world with scrye, teleport, speak with animals . . . etc. Just don't give away too much (See #2). Maybe the city was overrun, but the initial reports say its holding out, for example.
4. Rumors should come from odd sources. Speak With Animals is great for this. That chipmunk knows "Green pig faced 2-legs walked this way a sunrise ago." Writing on dungeon walls is good too (especially in other languages): "This passage is forbidden by Orcish Law." (written in goblin). Or a Magic Mouth reciting a cryptic verse: "When the last stone falls, Loamgrash will stand tall."
Rumors are a great tool for any DM to use. They are especially important in sandbox games:
1. They give choices to the players. Or at least the illusion of choice:
"The Citadel is overrun by goblins, again."
"In the Manors District, there is a powerful artifact."
"The Temples District was walled off, because of soul-sucking undead."
Chances are, a low level party will head to the Citadel, a mid-level party will head to the Temples, and a high level party will head to the Manors.
2. They allow for preparation against threats expected.
A low level party could handle the Temples with Lesser Restoration scrolls, and other protective magic. This would allow them to explore the area, and while more dangerous than goblins, they would still have a (slightly less than) reasonable chance of survival.
3. They allow the players to find what they want:
"Oh, you're searching for the Sword of Ultimate McGuffin, huh? I don't know where that is, but, I did hear about a flaming sword in a hydra's lair. I was told the adventurers that saw it got there through the sewers. Which way in the sewers? I didn't ask."
Someone wants a particular magic item? Send them in a direction. You wanted them to head toward that red dragon next session? Guess who was rumored to have it . . .
4. They create motivation.
"The Adamantine Thrusters of Justice are also looking for the Sword of Ultimate McGuffin. They said they were heading back into the dungeon tomorrow morning."
Rival adventuring parties, impending doom of a town, armies marching to war. Any of these things (and many others) will get your PCs off their butts and moving.
I could go on all day about why rumors are good, and how to keep them from being bad. Just use them. And use them well.