How does ASICMINER use the blockchain to record share ownership?
ASICMINER management (generally friedcat on Bitcointalk) keeps records of all ownership of shares in their database. The only way to confirm ownership is either through friedcat's acknowledgment or by receipt of dividend payments. Share owners' bitcoin addresses are tied to their shares for dividend payouts.

If a share owner wants to transfer his ownership to someone else, he must provide friedcat with the new bitcoin address. With auction sales, this is typically done through a third-party escrow service for the safety of both buyer and seller.