With lightning there's no history of the transaction put into a block. So for that, its quite private. However, if you're using custodial lightning like wallet of satoshi they could be logging all of your transactions. Running your own LN node and using unannounced channels is best, though more resourceful and more learning required.

LSP models like using LN on electrum or phoenix is a good solution for the time being.

As for liquid, it has confidential transactions so the amount is hidden. It is also kinda custodial and it can be linked when you peg in/out. I see it as a good way to put small change into from an onchain TX. It can also be used to refill a LN channel through boltz or another swap service.