What I worry about, is in countries like the US/UK where "having a student debt" has somehow become a badge of honour, I can see "selling a kidney" to get through university becoming the same honourable demonstration that you are committed to do what it take to succeed.
... and by extension, show people who are not as committed. You can see it coming: "my son sold his kidney to succeed, why should we give even more to the people that are not even ready to take such a small step".
In general, we seem to expect people to take an ever growing amount of risk for the reward of maintaining their social standing. Being able to sell your body in piece is just raising the bar even higher.
Middle men are easy to fix. Require the kidney transplant be done in a registered location and don't accept "nameless" kidneys.
So do the same failed arguments -- how has legalization and regulation of the market gone for Germany? Or rather how has it gone for German prostitutes, because from a tax revenue perspective Germany is doing wonderfully, with the perverse incentive to keep exploiting poor and trafficked women.
Luckily, someone is offering $1,000 for a kidney and you're compatible! You can fix your car / post your bail / pay the loan shark, with $500 left over!
You're pretty much forced to sell your organ at this point. And this point is reached all the time by the economically disadvantaged.
Of course, without the last resort of selling body parts they'd be even more screwed, so I don't know the right thing to implement in this case. Just wanted to answer your question.