Supposedly they're going to do a fair bit of it on device for privacy reasons, so the only payment for that will be RAM and battery power.

For stuff that can't be run on phones, some of it will be run on Apple's servers, which I'm assuming Apple is eating the cost of for the time being.

Stuff that needs heavy reasoning or external knowledge will be processed by google, in exchange for $1 billion a year. However Google already pays Apple $20 billion a year for google to be the default iOS search engine, so you could view this as just changing to google paying $19 billion a year instead.