I am not sure if I get your question right. Please tell me if not and I shall iterate on my answer.
First of all the lightning network indeed has a different privacy model only opening and closing of channels is stored to the redundant public ledger called blockchain. Once the channel is open even when routing is applied the payment itself is pretty private.
I have been told that people work on a truly decentralized marketplace on top of lightning. Obviously payment or exchanging of goods and currency seems like an obvious application.
In a similar direction I believe selling digital goods and file-sharing can be powered on top of lightning as it does not require the person who is shopping to register an account. Any digital good could be symmetrically encrypted with the preimage of the invoice as the key
I know that the people from bhb networks are trying to build the rgb protocol on top of lightning which would give colored coins and the ability to issue tokens which could also be transferred via lightning.
On a protocol level there is an ongoing effort to implement rendez vous routing which is an improvement over onion routing. With rendez vous the person getting paid will be able to hide the location of his/her private node it might maintain by proving the remainder of an onion with the issued invoice making a payment from some rendez vous point to the payee.
I personally suggested JIT routing which would produce less routing failures and thus keep channel balance private.
As mentioned I hope your question was meant to go in this direction.