Does a copy of a spell on the stack share targets with the original or can you choose new ones?

When a spell or ability creates a copy of a spell on the stack, the copy starts out with all the same choices as the original — including targets — but then the controller of the copy may choose new targets for it (CR 707.10). This is not optional in the sense that you must choose legal targets if the spell requires them, but you are free to pick different ones than the original or keep the same ones.

This most commonly comes up with effects like Twincast, Strionic Resonator (copying a triggered ability), or Fork effects. The copy is placed on the stack and its controller immediately gets a chance to select new targets before anyone receives priority (CR 707.10, 707.2).

It is important to note that the copy is an independent object on the stack. Once targets are chosen for the copy, it resolves separately from the original. Countering one does not counter the other, and each checks its own targets on resolution (CR 608.2b).

Example: Your opponent casts Lightning Bolt targeting your 3/3 creature. You cast Fork, copying the Bolt. The Fork copy goes on the stack; you may now choose a new target for the copy — say, your opponent's face — while the original Bolt still targets your creature. Both will resolve independently.