Right now you're the player that's undergeared. Imagine instead if you were the player overgeared and you wanted a vanity item. Everytime you played there was someone undergeared compared to you.

Would you want to run the dungeon over and over having no chance to get the loot you wanted? What's fun about that? If you each run the dungeon once, then you should each have a crack at the item. Need gives priority to the class you play, so if you want gear for a specific class, then play as that class.

Also, if you're able complete the dungeon without that gear, then you obviously don't need that gear to complete the same dungeon again. You don't "need" that gear to run the dungeon any more than the player who rolls for vanity.