My wife is a non-AT pilot who works for a major airport, I am an ex-pilot (maybe an again pilot, depending on medical).

This I know: AT pilots sleep in the cockpit, taking naps while the other pilot watches instruments. There may be exceptions, but I don't know any.

The work hours are insane. They were bad before the cut-backs, now they are terrible. It's a taboo topic, and pilots (who work for love more than money -- very few pilots are well-paid, and the education is very expensive) don't like to talk about it to non-pilots for obvious reasons. Nonetheless, there is a tacit understanding that sleeping in shifts is safer than trying to meet the impossible demands of the airlines. And both pilots are always awake at take-off and landing, the only points of most modern flights that are likely to demand emergency action.

I think it would be far better to have FAA (and international) guidelines that allow for real rest between legs and flight days than to formalise pilots sleeping in the cockpit. It is really much better to have two alert pilots than to have one alert one and one sleeping one, or to have two pilots who are barely functional. While take-off and landing are the most risky portions of a flight, it's good to have two trained minds and two pairs of expert eyes peeled for the random stuff (mostly weather-related) that happens mid-flight. Most legs are short enough that it is easily possible to remain alert throughout, provided you're well-rested when you begin it.

I know pilots who time their own flights as passengers to take advantage of this, trying to arrange their passage for a crew that is coming back from a long rest period. If that doesn't illustrate how serious the problem is, I don't know what does. But, it also suggests that the problem is one with a solution: let pilots actually sleep on the ground!