Organ theft

You had a few too many drinks in a disreputable bar someplace in an otherwise tropical paradise. You don't remember much of last night, only that you are in a bathtub full of ice, and you have some stitches along the sides of your back... is not how organ theft occurs. In the first world, it's (mostly) an urban legend, where a group of enterprising thieves steal people's organs and sell them to someone else.

However, this does not mean that organ theft is non-existent.

Deconstructing the legend
Organ donors and recipients need to be compatible; you need a donor of the same blood type, similar genes, similar size, and a whole slew of other requirements. It's also important that the organ comes from a relatively healthy donor, as there's not much point in trading kidney failure for AIDS. Organs are very much alive, and will not survive forever outside the human body; kidneys can last for about 36 hours, livers 12 hours, and hearts a mere 4, so generally speaking you have to be ready to have the surgery at the drop of a hat. Once the operation is done, your body will begin to attack the organ as if it was a foreign invader; to prevent this, immuno-suppressant drugs need to be taken for life. So how much does this normally cost? Well, organ transplants are one of the most expensive operations known to medicine, at anywhere from a quarter million to over a million dollars.

So you find yourself some person whom no one will miss. Now you have a big ol' pile of organs to remove and then sell. What do you have to do? The checklist:
 * 1) Find a matching patient
 * 2) Find one immediately
 * 3) Preferably, find multiple patients; don't let all that product go to waste!
 * 4) Have an adequately sterile location for practicing internal medicine
 * 5) Perform a complicated operation with qualified surgeons and support staff
 * 6) Hire qualified surgeons that somehow can't find more ethical work, but are still somehow ethical enough to not turn on you to save their own skin
 * 7) Do all this with the infrastructure and networks required without anyone blowing the whistle on you
 * 8) Afford the bribes/hired goons to do 'damage control' for when someone does find out about you (on the other hand, more organs!)
 * 9) Convince your customers that you actually have some quality control in spite of being a black-market organ dealer

On the flip side, if you need an organ, your options are...
 * 1) Wait on the list and hope for the best
 * 2) Spend the money to relocate to where the list is the shortest, e.g., Steve Jobs
 * 3) Use political connections or money to simply bribe your way to the top of the list, e.g., Dick Cheney
 * 4) Contact your local organized criminal gang, hope they have the very specific organ you need, are somehow capable of actually doing the surgery, and risk being forever blackmailed and extorted as an accessory to murder

This, of course, means that the organized criminal gang has to charge less — much, much less — than simply relocating or bribery. If the crime is going to occur, it'd be easier to simply infiltrate the supply chain and be the government officials who decide where donated organs go, which leads us to...

Reconstructing the legend
Organs do indeed go missing, but it's not of the random street gang variety. In each of these cases, it's much more organized and it involves either the government itself doing the harvesting or people being paid to donate organs.

Shortage of organ donors
Most countries have a larger demand for organs than supply. Some people have fears that if people check the "organ donor" box when they get a driver's license, if they are injured, their doctors will intentionally let them die to harvest their organs, which is not true. (In fact, in Portugal, one has to register to be a non-donor after death. By that logic, Portuguese doctors would be allowing a lot of deaths - and malpractice policies would be through the roof.) The result is that a number of people who would otherwise donate might be afraid to agree to become organ donors, which increases the suffering of those currently living. The scarcity of organs can push people to try to buy matching organs, often from somebody in the third world.

Once again, if these people have the money to buy the organs at the prices alleged, they have the money to simply bribe someone or move to get on a shorter waiting list.

Chinese prisoners
China sources a number of organs from recently executed prisoners. However, it's not clear if the prisoners ever agreed to donate their organs. Combined with secret/"random" executions, this leads to the not too far-fetched conspiracy theory that China schedules prisoner executions whenever someone rich/important needs an organ. That China has built numerous concentration camps in recent years should be scary enough, and it's not too far fetched an idea that some occasionally go missing.

A section on "organ pillaging" featured prominently on China Uncensored under the title of "China's Secret Holocaust", a show which was created by practitioners of the Falun Gong cult. Strangely, are conspicuously absent from the YouTube channel of China Uncensored, which instead focused primarily on prisoners of conscience who were victimized by this practice.

For years, China denied organ harvesting. Then, officially, China ended the practice of harvesting prisoner organs starting January 1st, 2015, which meant that until 2015, they were definitely harvesting organs. Even so, they are routinely accused of continuing to harvest organs from Falun Gong and Uyghurs.

Actual organ theft

 * Sorry, were you looking for Dick Cheney?

In most cases of real organ theft, a poor person agrees to sell one of their kidneys for, say, $10,000. The person willingly goes to a medical center, and the organ is removed. But after the operation, the person is only paid $1000 (if at all), or may not know what they agreed to. This is a breach of contract and horrible exploitation of people in absolute poverty, obviously, but is not the scenario people tend to imagine.

Not all of the people selling their kidneys are the victims (well, sort of); the hospitals and doctors are often defrauded in the process. Gangs would forge documents claiming that the organ was being donated to a relative, and the hospital would perform the operation while the gang would get a huge profit.

As the world improves, fewer people will be desperate enough to be at risk of these sorts of scams.

Hopefully.

Witch doctors
The trade in human body parts predates modern medicine by millennia. While today you might get the kidney of a biker to gain his urine filtration, warriors of yesteryear might eat the hearts of their enemies to gain their courage (but not their luck or skill in battle, presumably, since it's the losers' hearts being eaten).

If you made the unfortunate decision to be born an albino or pygmy (or an albino pygmy) in Africa, you could find that you are highly valued by your society. Literally: your body parts could be worth up to $75,000, more than what it'd cost to have someone murdered in the developed world, let alone a part of the world that's dirt poor. Really, while the alties may be a danger to themselves, at least their beliefs won't end up getting you killed. Then again....