It would start off with someone selling a kidney, maybe the recipients family fundraised, and the donor would have a reasonable price for it plus the warm glow of satisfaction that they had helped somebody. This state of affairs would last precisely until someone in America or Europe needed an organ urgently, either for themself or a loved one. Then somewhere in China, or India, maybe Brasil or one of the former Eastern bloc states, a teenage boy or girl, most likely someone poor, or even homeless goes missing and winds up dead. Their organ reaches it's new owner and some scumbag pockets a lot of cash at someone else's (considerable) cost.
Our governments can't fight drugs, can't fight people trafficking, can't curb gangs, or keep guns off the streets. They sure as hell wouldn't be able to prevent the abuses that would take place. Such things may well be occuring as we type, but legalising organ trade would blow the lid off it. At the moment if you want to use a stolen organ for a transplant you need the financial ability to find a surgeon willing to do the work with no questions asked, most likely out of the country. If the trade was legalised, every American and European on any donor list would want one ASAP, their insurance would cover the cost and you will have created a considerable demand. Law of the market means someone is going to cater to that demand.
It's a nice idea, but it doesn't bear any great deal of exploration. On a related note, isn't it a sad indictment of us that we can debate an issue that would so clearly exploit the poor around the world when so few of us actually carry donor cards ourselves?