And if they're giving password of Twitter account to support staff, how they rely on the staff?
In both cases, very few people handle the Twitter accounts in general, and usually they reveal some part of their identity. Some companies decide to create multiple customer service accounts for multiple people as well.
My guess is, they have some power-user applications running some searches for key words (obviously company names) and maybe, if they are really serious, some sentiment analysis (e.g. "Regions sucks!" is much more important than "headed to Regions then to lunch.")
The larger companies will hire a small team, or may add this to the responsibility list of someone who they feel takes initiative. Companies like MediaTemple have customer support folks who monitor this like any other channel, with a scheduled switchoff to other support reps. Small companies have someone doing this and usually many other tasks at the same time.
Long story short, one leg of my flight was cancelled and I instead took a bus. I was promised I would get a refund with 2 weeks, yada yada, and I end up not getting it. After 2-3 more phone calls and similar cycles, I get tired and post to them publicly on twitter and facebook.
Lo and behold, they got on the case immediately and I got my refund in the next day or two.
Is there a moral there though? Should one use twitter and/or facebook to give a company poor publicity and then have them specially take care of you? Unclear, but since this was a case where I had already tried to deal with them on the phone 3-4 times, I felt justified.