How does ASICMINER use the blockchain to record share ownership? My understanding is that ownership of ASICMINER shares (direct ownership, that is; not passthroughs) is recorded somehow in the blockchain, itself.

How does that work, exactly?  How can I confirm that a person claiming to have 300 shares available for sale actually owns 300 (or more) shares?   I also see people saying they "received their satoshis transaction" (in addition to their dividends payment).  I suspect that is somehow related, but that can hopefully be clarified in an answer as well.
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.