Who gets the copies from Eye of the Storm when control of a spell is switched before exiling the spell?
The player who cast original the spell controls the copies of the exiled spells.

The full text of Eye of the Storm's ability says


  Whenever a player casts an instant or sorcery card, exile it. Then that player copies each instant or sorcery card exiled with Eye of the Storm. For each copy, the player may cast the copy without paying its mana cost.


There is only one possible antecedent for the phrase "that player" in that ability.