I was curious if we could get any more information about off-the-chain anonymous transactions? 

If you would want to send a transaction off-the-chain, wouldn't that mean that you'd already have to have the coins in some storage held by a third party?  That is, if I want to send 100 coins from my wallet to account X off-the-chain, then the public ledger would still have me as the owner of those 100 coins, and therefore, I could doublespend my coins.  However, if my coins were not held in my wallet, but in a third party storage, then they would be accountable for validating this transaction without the blockchain.

Is the above thinking correct or is there another way to allow off-the-chain transactions?