Permission Denied's effect applies to all opponents, not just the player who controlled the spell targeted by Permission Denied.

If Permission Denied resolves but the spell isn't countered (perhaps because it can't be countered), opponents will still be unable to cast noncreature spells this turn.

If the spell is an illegal target as Permission Denied tries to resolve, it won't resolve and none of its effects will happen. Your opponents will still be able to cast noncreature spells this turn.