With Shroud, that permanent or player cannot be the target of any spell or ability that targets. Hexproof is the same, except only with spells or abilities your opponents control.
So say you have an Invisible Stalker out, you can cast Giant Growth on it, but your opponent can't Murder it since Murder targets. However, spells or abilities that don't target, like Supreme Verdict , can still hit them.
Both of you are wrong.
Shroud means This object can't be the target of spells or abilities. Nothing can target an object with shroud, but nontarget effects (e.g. Day of Judgment , overloaded Mizzium Mortars ) can still affect shrouded objects.
Hexproof means This object can't be the target of spells or abilities your opponents control. The "your" refers to the controller of the hexproof object. The controller of a hexproof object (and any of the teammates of that controller) can still target a hexproof object. However, none of the controller's opponents can target it. Nontarget effects can still affect hexproof objects regardless of who controls them.