Considerations of privacy when using Lightning versus CoinJoins

Ring signature transactions (Monero) and CoinJoin transactions (Wasabi / Samourai) give you algorithmically-enhanced privacy.  The input funds are mixed with others' funds, and the outputs can technically be very hard or impossible to deanonymize.

The privacy properties of Lightning are different.  They are not algorithmically determined — rather, they are a practical consequence of how the Lightning network works.

What I mean by practical privacy, is that the State has no easy way to peer into what the channels' messages (sends/receives) are, because Lightning nodes and routes are decentralized, and the communications between nodes are encrypted.  All the State sees is the settlement transactions (opening/closing channels) which are always between two peers and look like "I sent you X bitcoin, and two years from now you returned X-Y bitcoin to this other address".

Unless on-chain transactions were to become de facto illegal or suspicious — a potential development — you have much more privacy when spending via Lightning than when spending on-chain — some would say almost equivalent in practice to Monero or CoinJoin transactions.  This assumes, of course, that the decentralization property of the Lightning network continues or improves over time, rather than converging into a few nodes that the State can squeeze with its customary physical aggression or threats.  If the network coalesces around a few custodians, then the State can put them all in prison, and that's the end of Lightning and its privacy.

This is why decentralization is key.