By the rules
In The Dark Eye, there is Streetwise (Gassenwissen in german, page 196 in the german rules) for this. In the rules, there is a list of things you can do with this skill. Gather Rumors is included (and has a general modifier of +1 for the skill check)
There is also a list of outcomes for a streetwise check:
- On a failed check:
The hero does not gather any helpful information
The hero gathers useful information. The better the result, the more useful the information he can gather.
- Critical Success:
The hero gathers more information than he expected/wanted or finds a special person that can help him for a better price or something similar.
- Critical Failure:
The hero is ambushed by some bandits who want to rob him
Do the players roll the check?
The answer is yes. In general, the players in TDE roll their hero's skill checks. They know what will happen if they fail or succeed anyway. Also, failing on the roll will not result in them getting wrong information, but instead in them getting no information or unimportant information. So, a hero looking for the hideout of the bandits in town will not get the wrong position on a failure, but instead no information at all. Maybe all he gets is stuff like "Oh, did you hear, Queen Rohaja is going to marry the prince of Albernia!" and similar useless stuff.
How I would handle it
To be honest, the idea of "no wrong rumours" or "on every single critical failure, bandits jump the hero" is boring. Sure, that is what the rules say, so if you want a clear, by the book, ruling, stop reading here.
Handling wrong rumours
Wrong rumours require your players to be able to handle metagaming issues and really differentiate between player knowledge and character knowledge. If you search a bit on this stack, you can find many questions that handle this, so I won't repeat that here, as it is not really the core of your question. If you try it and you players can't handle it, then don't give out false information.
Handling critical failures
This is simple, really. Take the bandit ambush that the rules say will happen as inspiration. The hero tried to find information about the thieves guild? Well, they notice and suspect him to be with the guard, so they become extra cautious or even actively try to stop him. He tried to find someone who sells poison? The guard notices and questions the hero.
Vary it up, and it will give you an opportunity to give your players new enemies, new contacts or new difficulties they have to face.
(Btw, according to the rules, the bandit ambush will happen no matter what the hero wanted to use Streetwise for. Searching a tavern, gathering rumours, find a buyer for a stolen artifact... you name it.)