How can "undo" bitcoin transactions be "private"?
The failed transaction won't get stored in a block on the blockchain, so it will be hard for an average person to find them.  I think most of the Bitcoin nodes will delete them from memory eventually.

Also, these sneaky undo transactions are probably not broadcast so publicly in the same way that normal transactions are; they are probably sent to a smaller network of miners who want to participate in this undo system.