How do people smuggle stuff with private airplanes?
There are lots of ways to smuggle goods with planes and it depends what you are smuggling and the aircraft on which you are smuggling it. Smugglers often bank on the fact that most countries do not have or allocate the resources to search every aircraft throughly enough from top to bottom before every flight as that is simply impractical on some level. Some times all you need is a good audio equipment road case. 

Generally you need to get past customs which happens on the ground and is its own set of issues. In the USA the Customs And Border Protection takes care of a lot of the searching. Smuggling sometimes happens on commercial aircraft and there is plenty of info on it here on the CBP website. In this case the goods are smuggled on aircraft that are very much on radar and flying at assigned ATC speeds. Sometimes they are even loaded by official personal. On occasion even the head honcho is involved, moving some strange things. 

It is possible to "fly under the radar" but that also depends on where you are going and what their radar coverage is. Steps have been taken over the years to combat this issue. It seems that some of the smugglers have once again resorted to small aircraft.

Barry Seal, the inspiration for the movie American Made was historically a pretty prolific smuggler. The movie takes some poetic license but is based on some facts. 


  Seal used his knowledge of aviation to become the infamous smuggler he
  was. Once in U.S. airspace, Seal would drop to 500 feet and slowed to
  120 knots to mimic, on radar screens, helicopters which frequently
  flew from oil rigs to the coast.
  
  Within U.S. airspace, Seal would have people on the ground monitor for
  any signs his planes were being tailed. If they were, the mission was
  aborted. If not, they would continue on to drop sites over the
  Louisiana bayou, where duffel bags full of cocaine were dropped into
  the swamp. Helicopters would pick up the contraband and take them to
  off-loading sites, and then on to Ochoa distributors in Miami by car
  or truck.


Smugglers are still attempting to move product in older, but fast, biz jets with fake or missing registrations. To avoid being discovered they are apparently destroying the planes after they use them sometimes. 

There is a great article on some of the history of air smuggling here. Which dates back to the early days of aviation.

This is a fairly interesting story...



Just remember, aviation smuggling is Illegal very illegal