Paying headhunters to place you

I'll start by saying I'd never do this, but has anyone wondered why people don't offer to pay headhunters to place them into jobs? It seems like that would be a good way for HHs to supplement their income. The only thing I could think of is that that's a conflict of interest and no PE shop would work with a HH that also takes money from a candidate. At the same time, PE shops themselves ultimately make the call on who they want to interview, which is the impartiality check in and of itself. Thoughts?