Honestly, it would depend on who comprised the pair.
Holmes and Moriarty, for example, would have a fascinating intellectual conversation about using one’s genius for good or selfish purposes.
Dracula and VanHelsing would result in a blood bath. I’m unsure who would ultimately lose out, though.
Mr. Spock vs. Evil Spock (with a goatee! For villainy!) or Cpt. Kirk vs. Evil Kirk would definitely be a long brawl just like in TOS.
James Bond and just about anyone else…well, there would be a short martial arts duel that would end with either an unconscious male villain or one very satisfied lady villain.