The FAA is the Federal Aviation Administration which is part of the Federal Government which makes FAA controllers government employees. There are contractors (Midwest ATC, RVA, Serco) which provide ATC services at some of our smaller VFR airports.
The sleeping problem arises from the schedule that controllers work, which is often consist of rotating shifts nights, days, and mids. Often you only have 8 hours off between shifts, so when you figure in commuting to/from home and work, spending time with your family, you may be only getting 3 to 6 hours sleep between shifts. I've had schedules where I worked a 3 night shifts, a day shift (quick turn) then back that night for a midnight shift (another quick turn), so you can imagine how tired I would be.
With this latest incident in Miami, I wonder why the other controller didn't wake his co-worker up so he could had requested a break instead of calling a supervisor or manager to come and catch his co-worker sleeping. This stabbing your fellow controller in the back just leads to more problems.