The Drow Elite Warrior does 3d6 extra poison damage (no save!), it's not a magical sword but an ability of the creature. So that same sword in the hands of another character is just a sword, it's the super special drow elite poison that makes the damage. I'd feel cheated if I were a player, considering that according to the DMG errata injure poisons stop working after being delivered through a wound.
Injury poison can be applied to weapons, ammunition, trap components, and other objects that deal piercing or slashing damage and remains potent until delivered through a wound or washed off.
The rules say nothing about this, that I could find. Any suggestions about how to handle it in a game, particularly while playing Out of the Abyss? If they by some strike of luck manage to kill or subdue an Elite, they are going to want to have that super-powerful sword. Or the first time they poison their swords they are going to be annoyed that the poison just works for one attack... Probably they won't notice, but I rather be prepared.