From what ive seen privacy with lightning is increasing with Bolt12 with blind tx. depending on how you use a wallet (custodial vs non) can be a leak. Also currently receiver of sats can be seen. Its not perfect but there are upgrades being worked on.
Amethyst lets you send Sats anonymously.
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Corrext, sender is anonymous but someone looking to track sats can see if a wallet received them. Thats the privacy being worked on
