"Productively" means people are doing something more with it than tweeting "LOL! OMG that episode of the Hills was soooo Hillarious!"... like you are. Aren't you ashamed.

Student 'Twitters' His Way out of Egyptian Jail
from CNN.com
Did you know that Twitter can help save your ass from political persecution in a foreign country? When he was arrested while photographing anti-government protests in Egypt, University of California-Berkley grad student James Buck sent a one-word tweet to twitter: "Arrested." That single word set in motion a process that led to his eventual release. Too bad you can't say the same for his translator...

Internet activism: Twitter as a tool to promote homeless issues
Kevin Barbieux gives you glimpses into the word of homelessness and reminds us that homeless people aren't some sort of strange untouchable caste-- they're us. "Chronically homeless" Barbieux didn't have a place to lay his head at night, but managed to frequently update his blog and Twitter posts using laptops donated to him by friends. And if right now, you're thinking, "What's a homeless guy gonna do with a laptop? Shouldn't he be buying FOOD with that money instead?", I invite you to read and internalize the slogan for Barbieux's blog: "There is more to homeless people than being homeless."

Read my Tweets! Lots of New Taxes!
Earlier this year, a bunch of people with too much time on their hands tried to make a big deal out of the fact that while Barack Obama was following as many Twitterers as were following him, Hillary Clinton was following 0 people while talking to thousands. Honestly, as if Barack pays attention to any of your "Just ate TWO whole moonpies by myself!" tweets.
Still, it's interesting that they're making use of the technology at all, even if their pages read more like an electronic schedule (" In McKeesport & Pittsburgh, PA today finishing the "On Track for Change" tour and reminding everyone in PA to vote tomorrow!") and less like the highly personable blog medium it's meant to be. I'd like to see more personal messages, direct from the candidates' cellphones. Something like, "Mike's in Philly has the BEST cheese steak EVAR!"
As it stands, a more interesting alternative is Politweets, which tracks and categorizes any mention of the current political candidates made by regular posters in the Twitter-sphere.