You can see the cards that are exiled with Jester's Scepter, but no one else can.

Because the cards that are exiled are not hidden to you, you can choose which one to put into its owner's graveyard when you activate the second ability.

When Jester's Scepter's second ability is activated, any of the exiled cards may be put into its owner's graveyard, and any spell may be targeted, whether or not the card and the spell have the same name. If they don't have the same name, the effect does nothing. The spell's controller and the owner of the card put into a graveyard this way may be different.

If half of a split card is cast (for example, Hit), and you put one of those split cards into its owner's graveyard with Jester's Scepter (for example, Hit/Run), the spell is countered.