Dear James,
Yes, over the years I have read (at this site and elsewhere) of successful "swaps" arranged through those sites. But I'm sorry that I can't give you any hard data on percentage of success.
I have also known a surprising number of cases where once the need was known a quite unsuspected volunteer came forward, sometimes a casual acquaintance or neighbor who was not really close to the recipient. So it is worth getting out the information about the need and the possibility, even though it means losing some privacy. I know personally of several people who decided to donate a kidney when an announcement about the need was made at their church or synagogue, even though they did not even know the name of the person.
With regard to your original question: as far as I know Iran is the only country where it is entirely legal to give up an organ for transplant in return for money, but it is only legal there between Iranian citizens, not for foreigners. Their system appears to work. The government agency supervises the transaction to try to guarantee that the donor is healthy enough to donate safely and to determine the payment amount. They no longer have a waiting list for kidneys. But again, it is only among for their own citizens.
There are other countries where the sale of organs is illegal, but the government looks the other way when the organ is sold but the seller pretends to be a friend or distant relative of the recipient, so the paper-work is done even though everyone knows it is illegal. Other countries do organ sales and surgeries entirely illegally, paying off the authorities.
As Karol noted, a big danger in using overseas transplants is that the brokers may not ensure that the sellers are actually healthy enough to donate safely and are free of diseases that are passed on with the transplanted organ. There is likewise the danger that the seller will not even be paid the promised money by the broker after the surgery, nor get proper post-operative care.
Although mainland China has officially outlawed the practice, it appears that they continue to sell the organs of executed prisoners to foreign buyers, and will even schedule the execution for the convenience of the buyer with enough money.
So while your concern for your son is understandable, you would have a lot to think about before opting to try a foreign transplant.
I forget to mention another option in the U.S which is the "list exchange" which one or two people at this site have used. If there is a willing but incompatible donor some hospitals will allow that donor to give the kidney to whichever patient the hospital chooses from the waiting list. In return that donor's relative is moved to the top of the waiting list for a kidney from a DECEASED donor thus greatly reducing the wait time. (Note: kidneys from deceased donors, on average, do not last as long as kidneys from living donors.)
Another option, which I am less familiar with, is
www.matchingdonors.com . Here persons willing do donate look over postings from persons in need of a transplant and may choose one to donate to. (Hospitals may take precautions in such cases to try to make sure that no money has been exchanged under the table.) I believe there is a fee involved to post at that site. They have arranged some successful transplants, but I don't know how many.
I apologize if I am dumping too much information on you, but you should also know that while in the U.S. it is illegal to buy or sell an organ for transplant, it IS legal for the recipient to reimburse the donor for ALL financial losses suffered because of the transplant (lost wages during surgery and weeks/months of recovery, travel/lodging expenses for testing and surgery, etc.) The donor does not GAIN any money, but is reimbursed so as not to LOSE any money. There are willing donors who are unable to donate unless someone covers their lost wages, expenses, etc. So if you are able to do so it would be wise in your search postings to explain that you would reimburse the donor for all lost wages and other expenses.
best wishes,
Fr. Pat