There are way too many low quality (twitter?) messages. Personally, I would not spend any of my time refuting and upvoting them. People have only 24 hour in a day, you need to provide better links, so people not only filter the data, but find new interesting stuff.
There's not really any good way to say "show me only good Twitter stuff related to this content." The top of the page should show you the best stuff, and the live streams will be for more hardcore users, kind of like the people who scan through the "new" page on HN. But we are definitely trying to find way to make that better if you have any suggestions.
Also, part of our thesis as to how that works is that information can be "important" even if it's false - it's important to mark it out as false, since it's going to be spreading regardless. So in theory the two actions should be separate, and the incorrect information should be marked as such. Could be wrong on that - we'll see.