Alby is a custodial wallet so they can very easily see when one npub zaps another npub. Not very private. Also, zaps are usually published as nostr events so that clients can show the zap tally. Again, not private. If you want to send someone BTC privately via nostr, better to DM some ecash (look up eNuts, Minibits apps, both using Cashu protocol). Still custodial but private.

Typical lightning payments (where a user generates an invoice and another user pays it) are pretty good for privacy, especially for the sender (I believe). Others can correct me here.

As an aside, based your profile pic you may be based in Toronto. If so, hit up nostr:npub1826v365he5ty69lk3xgvzqrwy8587vdfrxnsz0k09khzustf8r7s6j7t95 for a torontoplebs.ca NIP-05 verification handle.

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Hey Ben, thank you for replying! And yes - I am a Torontonian, so thanks for letting me know lol.

No problem! I grew up in the gta, now live down the road in London.

There's a coffee shop on st. Clair west that accepts bitcoin for payment if you ever want to use bitcoin in person lol.

Oh nice - didn't know. I might check it out one day

Would a lightning invoice (not a nostr zap but a regular invoice) also give the information of the sender?

I believe the only potential identifying information would be the IP address where the invoice is generated. Thus better privacy for the sender than the recipient. Here’s a pretty good article that summarizes lightning privacy and the various ways one can try to figure out who paid whom. https://voltage.cloud/blog/lightning-network-faq/lightning-network-privacy-explainer/