Most of the companies doing customer support via Twitter are either very small (as in, 20-100 people) or very large (as in major banking companies).

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.