The primary mistake most job hunters make is desperation. Tweeting out public pleas for help is unlikely to earn you a job. Instead, tweet out things that are likely to impress potential employers. Keep your tweets clean, humorous, authoritative, non-political and responsible. Don't tweet anything you would not want a potential employer to know under your real name or job-hunting name. Use an alias to tweet out anatomical descriptions of the stripper at last night's bachelor party or post a picture of the tattoo you got in a Mexican prison after you were caught trying to smuggle drugs when you were 20.
Assume, correctly, that any potential employer you hook up with on Twitter will read everything you've tweeted before offering a job. In this economy there are often thousands of candidates for every position, and employers are using information gathered from social sites to weed out party animals, slackers, negative attitudes and morons. Before, during and after a Twitter job search, tweet responsibly.