Context here is to make a bot that can react to mentions. The bot should basically be listening to all relays, in case someone mentions it. And if it's mentioned, it should post back to the relays where it saw the mention note?

#[0]

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

NIP-65 compliant clients post to the mentioned person's "read" ("inbox") relays so that they will get the note, but only if that person has published at kind-10002 event with read relays specified in it.