Isn't there a wight attack in a barrow just outside the Shire in The Fellowship? It's not in the movie due to pacing, but as I recall the hobbits make the mistake of making camp in an old barrow and get attacked there.
I prefer my wights to be unarmed, but if I did use them with weapons at all, I wouldn't hesitate to allow them to channel the attack through the weapon (as some sort of undead attunement to the item, so they couldn't do it with just any blade, nor could the PCs benefit if they picked them up later).