I think you'd work it into the contract with the hiring company to get a % cut of each paycheck. The hiring companies probably already have a system in place for paying contract houses, this would just be a slight change. That'd probably be the easiest for the developer, too. And the 10% was just a number, you'd probably have to charge less than that. For example, the NBA limits agents to 4% (or so my quick Googling says), which seems like a reasonable fee.

If you start with just a few people you know well, I can see this being a good side job until the client list grows. The hard part is probably going to be that the people you know best are friends, friends may not want to pay you for your help.