In my understanding a lot of securitisation involves not actually selling the debt in the sense that the borrower is now paying someone different, but selling an additional instrument whose obligations are defined in terms of the original debt. So the originator has an obligation to pay on what the debtor pays back, but the obligation is canceled if the debtor does not pay. So the risk is transfered to the buyer of the debt-backed security. That's in its simplest form; generally debt-backed securities will amalgamate many original debts, and the aggregated risks can be sliced up and moved around in various ways depending on the terms of the security.
Cheers, Mike scandalum.wordpress.com