Do they have a team of people sitting around watching a Twitter feed, so that if anyone mentions Southwest they can pounce?
Actually, yes, they do.
I once tweeted to complain that of the four Southwest flights I took, a single one managed to get me to my destination on time. Every other flight was late in some way. My "favorite" of that group was the flight that landed 20 minutes ahead of schedule, only to be refused a gate at the airport and had to sit around on the taxiway somewhere for 40 minutes before being assigned a gate. (Apparently Southwest doesn't rent enough gates for all their flights at Seatac.) This counts as an "early" flight as far as their metrics are concerned, despite the fact that everyone was stuck on the plane until 20 minutes after it was scheduled to arrive.
Second place goes to the flight which landed at a Southwest hub that was stuck on the taxiway because there was no ground crew available to bring the plane to the gate and connect the jetway. Again: at a Southwest hub airport.
So, in any case, I tweeted this using Southwest (intentionally not using @SWA because I didn't really care at that point since by then I was done traveling) and got a response from a Southwest customer service agent.
The answer is yes: they do, in fact, search Twitter looking for people talking about Southwest and will reply to complaints.
Other businesses do this too. I've actually managed to get tech support issues resolved by whining about them on Twitter without even mentioning the a company handle. (For example, after complaining that I couldn't find drivers for Windows 8.1 for my Samsung laptop, a Samsung customer service agent replied telling me how to use their update tool to download working Windows 8 drivers.)