If you cast the spell and choose not to copy it, no opponent will get to copy it either.

If the spell requires targets, you choose the target of the original spell as you cast it. If you create a copy of the spell, you may choose new targets for the copy as you create that copy. Similarly, the opponent you chose to create a copy may choose new targets for that copy as it’s created. In other words, your opponent will know the targets of your original spell and your copy when choosing the new targets, if any, for their copy.

You choose whether to make a copy as the demonstrate ability resolves. This happens before the original spell resolves. Your copy goes on the stack above the original spell.

This means that if you cast a spell with demonstrate and both you and an opponent copy it, the opponent’s copy will resolve first, then your copy will resolve, and finally the original spell will resolve.