Applying shape keys to multiple objects

I want to make my character’s eyes be able to dilate, but also keep the eyes as separate entities. How do I apply the same shape key to those two eyes?

You create one eye, and create a shape key for it. You then mirror (or duplicate shift d s x -1) and (make sure origin is in the center of the mirrored eye) now you have two objects with identical shape keys. If you want them to be able to dilate separately you use a driver on each one, (or manually keyframe the shape key strength) if you want them synched up you would probably use a driver, driven by a single dedicated non-deform bone’s local rotation.