This is a fantastic question. I'm on Twitter, and I like it a lot. I'm able to connect with blog readers, follow my favorite screenwriters, and see headlines from my news outlets as stories are reported. I've even become "Twitter friends" with some industry people I've later met in real life.
Still, like all internet fun, Twitter can be a distraction. Whenever I hit a tough spot in a script I'm writing or get bored by a script I have to write coverage on, it's easy to click over to Twitter and procrastinate. Nora Ephron was definitely on to something when she suggested temporarily blocking your internet access to focus on your work.
Also, many writers fail to think about the consequences of what they Tweet. If you write that you hate a certain movie, you've permanently stamped it on the internet. What if you meet that movie's producer for a meeting someday? Would you want him/her to Google your name and see this insult pop up in the results? (I've blogged before about being aware of your internet presence - and the related question, "Should I start a blog?".) I'm sure this doesn't happen that often, and I'm not saying we should be super paranoid and self-censoring, but I try to ask myself if anything good will come from something I've Tweeted/Facebooked/blogged/etc. Is it worth it? Also, since I'm a new writer, I don't feel like I really have the authority to be bashing things (it's a little different when a screenwriter with 20 produced credits sends off an opinionated tweet). I'm not saying you shouldn't have opinions - just think before you put them all on the internet.
In terms of getting "discovered" on Twitter, it can happen - mostly for comedy writers. Many up-and-coming stand-up comedians gain popularity by tweeting hilarious one-liners. (Just make sure you save some jokes for your routines and scripts.) Also, you might consider tweeting in the voice of a specific "character" or concept in the hopes that it could become a show. CBS' now-defunct sitcom Sh** My Dad Says began with a Twitter account - and CBS also bought scripts based on Twitter accounts Dear Girls Above Me and Shh Don't Tell Steve. I think the key with this stuff is focus and specificity. These people aren't tweeting "OMG delicious breakfast! Time to work on my script!," you know? All their tweets are through a specific lens and concept. Also, for what it's worth, I don't think there's a drama equivalent of turning a Twitter account into a show (feel free to comment if you know otherwise).
You'll have to decide if you want to use your real name or stay anonymous. I use my name since I want people to be able to find me - but obviously, all my Tweets are then connected to me. If you're tweeting anonymously or under a "character," there's a little bit of a barrier in connecting your account to you as a writer - but I suppose if people loved your concept and wanted to contact you, they could send you a direct message (which isn't for public viewing). If you want to have more than one Twitter account, you can - you just need additional email addresses to open the accounts.
There aren't any official DOs and DON'Ts of Twitter, but here's my advice:
DO tweet things that you think other people will actually find interesting
DON'T tweet anything you wouldn't want people to find in a Google search for your name
DON'T tweet so often that you're super annoying
DON'T be afraid to tweet at your favorite writers, producers, etc. - but immediately asking, "will you read my script??" is probably not going to win them over
Lastly...perhaps this seems obvious, but don't think of Twitter as an easy way to skip the process of studying shows and writing good scripts. Twitter might help get you noticed, but I doubt you'll get very far without some solid writing samples.