How can a "subject to" property be sold without paying off the Deed first?

[Jeremy Jones]

Today I listened to (and very much enjoyed) Podcast #70 today which focused on buying properties "subject to".

One aspect that I feel unclear about: How can a property be sold without satisfying the existing mortgage Deed of Trust which would be attached to the Title?  Don't the lien-holders have to be satisfied in order for the a property to change hands?

For example, if I get a hard money loan on a property and they have 1st Position, I thought there was no way I could sell the property without satisfying their loan first (and thus they have secure collateral for their loan).  But the idea of buying a property "subject to" and closing using a real estate lawyer (not a Title company) seems to imply that a property can be sold without satisfying the underlying lien, and without contacting or involving the lienholders at all.

Can anyone help explain the mechanics of how the ownership can change hands without paying off an existing lien?  Usually after setting up a Purchase & Sale agreement it seems like the Title company immediately runs their title report and starts inquiring about any existing Liens and requests payoff information so they can be satisfied before filing a new Statutory Warranty Deed which changes the ownership of the property.