The note is only pushed to the relays that you’re writing to.

As for the person who was mentioned, they’re only getting a notification of the mention if they’re connected to one of the relays that the note was written to.

However, if the person mentioned is using a client that NIP-65/Gossip protocol, I’m not sure of the answer there. #[2]​ #[3]​ would be better to ask there

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So, if I want to make a bot react to mentions, it should connect to as many relays as possible as most clients don't do gossip yet.

Either that or query a NIP-50 relay that allows search like wss://search.nos.today