I'm investigating the possibility of going from a 51mm offset fork to a 42mm one, and in order to save on shipping cost (hopefully I'd find someone to trade with on the used market), I thought of whether I could only ship the relevant portion of the fork. Between the two versions of the fork in question (2020 RockShox Pike Ultimate 29" B4), what actually changes?
A fork's offset, or fork rake, is the distance between the front axle and the steering axis – the imaginary line running straight through the midpoint of the steerer tube. A fork's offset affects bike handling since it is a key factor in defining a bike's trail, which is the horizontal distance between the front tire's contact patch with the ground and the steering axis. Thus, the handling characteristics of a bike, specifically how the steering "feels" at any given speed or cornering situation, can be modified by changing the fork offset.
Since offset is defined by an imaginary line through the center of the steer tube and it's relationship (distance) to the front axle, changing where the steer tube is in this relationship directly changes the offset. To answer your question, the offset of a suspension fork like the RockShox Pike is determined at the CSU, crown-stanchion unit, or uppers. Changing the offset in a RockShox Pike is accomplished by switching the CSU, or uppers. The photo below shows 3 fork CSU's with varying amounts of offset.
Bike Radar has an interesting article regarding fork offset. The author experiments with different fork offsets. He accomplishes the switch to a different offset by simply changing his fork's CSU (uppers) to one with a different offset.