Correct me if I'm wrong, but doesn't the ability to rebroadcast a note to any relay from any client break the privacy offered by a relay?

Like if I share private information on a private relay to my family but one of the tech illiterate ones tap the broadcast button on #Amethyst, then the private data leaks to other relays right?

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Yes, that is very likely to happen.

However, privacy is not a binary thing. There are benefits to hiding most of your stuff while accepting that leaks will happen here and there.

That's why we designed the new DM protocol in such a way that even if the user re-broadcasts the outer or the inner event, it would still not leak any message or metadata information.

Because rebroadcasts will happen.