To the best of my knowledge, when a weapon has enhanced damage, it only applies to that weapon, not the other one if dual wielding and not anything that doesn’t use weapon damage. But what about other items with the enhanced damage affix? What does this affect? Main hand? both hands? something else?
Damage should be added to both hands.
If you put an Enhanced Damage Jewel into a weapon, it is only applied to that weapon. If you put it in your Shield, Armor, or Helmet, it will be applied to any weapon you equip, including off-hand.