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Wonder trade is sending out a pokemon of your choice and getting a pokemon someone else sent out. No means of communicating with each other prior or during, just straight up "I HOPE SOME ASSHOLE IN JAPAN WANTS A PIKACHU, 'CAUSE HE'S GETTING ONE."
Pokemon Amie is totally optional, but a max-happiness pokemon gets a lot of benefits (boosted XP, boosted crit rate, sometimes shrugs off status conditions, etc). These are in-game only benefits, so don't worry about getting haxxed by people online.
