What's the better character card deck set, Thief or Pirate?

I assume you can't use a card for the same character you are playing on, or am I mistaken?

You are mistaken.

But, I would go for a full pirate set, the pirate cards are more useful, especially if you use a mech rather than a corsair, than the thief cards, which give jump, speed, and meso rate, though DB's card does give % avoid, which could be useful.