I would at least say in the context of ZAPs it is not private at all due to the fact that it's a lightning payment that attaches nostr public key data to the payment it is a public record of who sent money to who.
Lightning privacy is complicated. The receiving side still has a lot of work to do in terms of privacy. Sending is a little bit more private, but it's also by no means perfect.
I'm not exactly sure how amethyst's private zaps work, so I guess there is that, but I'm Not sure if any other client has implemented private zapping.
So a Zap contains public information, but what about a regular transaction. Like if I open alby, enter someones lightning address and send some sats.
Again, without going into technical detail, it is not totally private, but that is better in terms of privacy than a Zap.
Although all the problems of using a custodian apply. Someone sending to you might only realize that they are sending to an Alny account, which I guess is fine, but Alby knows your account balances and who you're sending to and receiving from.
