The Lightning Network moves payments off the blockchain into payment channels. That changes the privacy picture, mostly for the better.
Strengths
- Not on the blockchain: individual payments are not published; only channel openings and closings are.
- Onion routing (BOLT 4): the payment is wrapped in layers. Each node in the route knows only the previous and next hop, not the whole route or its own position in it.
- Route blinding: the receiver can hide the last part of the route, so the payer does not learn the receiver's node.
Weaknesses
- Channel opens are on-chain and can be linked to your other coins.
- Public channels are announced to the network with their capacity.
- Invoices can reveal the receiver's node unless blinded paths are used.
- Custodial Lightning wallets see all your payments.
- Timing and amount correlation by nodes that sit on several points of the route.
Bitcoin's answer in practice
- Use a non-custodial wallet; if possible your own node.
- Prefer private (unannounced) channels for a personal wallet.
- Open channels with coins that are not linked to your identity, using coin control.
Lightning is often more private than on-chain payments for everyday use, but it is not anonymous by default.