The US government tracks statistics on this....Southwest's involuntary denied boarding rate is super low (.99 per 10,000). They are VERY GOOD at knowing exactly how many people will and will not show up for a particular flight.
They do overbook, but anytime I've been on an overbooked flight and they're concerned about running out of seats they ask for volunteers first. If you volunteer to take a later flight and they have to use your seat they typically refund what you paid for that flight, get you booked for the next one, and give you a credit for future travel. Depending on whether they get any takers, the offers tend to increase in value.
On one flight that was the last one for the night I ended up volunteering with 2 other passengers. They refunded our tickets, got us free hotel rooms (with shuttle to & from) and gave us credit for future flights. There was definitely no screaming :).
In all the traveling I've done with southwest I'd say only about 10-15% of flights with them have even been ones where they asked for volunteers, less than half of those actually ended up needing the volunteer's seats, and I've never been on a flight where a person (other than a crewmember flying standby) had to involuntarily give up a seat.
I'm shocked that nobody on that United flight on the news was willing to volunteer their seat for money...but I guess if several hundred bucks was a big deal to those passengers they wouldn't be flying United in the first place :P.
