It's not actually public access to "Data sets" - these folks are intercepting over the air live transponder data from each aircraft and then plotting the resulting information.
There are two primary forms of these types of transponders, one basic which just transmits the aircraft altitude and unique identifier, and the other advanced which transmits location, altitude, heading, intent (lading, taking off, descending etc).
In the United States, most aircraft only use the basic version, so actual plotting all aircraft using these interception techniques is much more difficult. However, any international aircraft that fly in the US, or aircraft that are international capable, can be tracked.
There are out of the box commercially available receivers and software that you can use to get started... but again be warned that until US adoption becomes more widespread you'll only be able to track International flights.
The code for a the basic (mode A/C) transponder isn't actually unique. It's just 4 base-8 digits settable by the pilot, with several codes reserved for specific uses. When on an ATC-assigned code, that code in only "unique" within a specific area. I might be assigned 0234 flying in Boston Center's airspace and someone else might be on 0234 in Phoenix.
Such services like http://radar.zhaw.ch used to have a 15 min delay one year ago for security reasons. Not sure if that is still true, it seems like it's realtime now.
Passur uses active radar and restricts access to its' apps outside US AFAIK.
Flightaware.com uses FAA feed (with a slight delay), but thus has full coverage of US airspace and a lot of other useful information (flight plans etc), but its' map sucks.
While sitting on a plane over the holidays, I was thinking something like this would be pretty cool, especially if Google Earth was the visualization engine.
flightstats.com seems to have flight tracking for a particular flight. I'm not sure if this isn't just a display of "elapsed flight time vs. scheduled arrival" though.
