In what circumstances is a calling card dropped on kill?

Solution 1:

  1. Any weapon will do.
  2. Yes. Looks like calling cards won't aid in Spy-checking after all.

I tested with the Gas Jockey item set. Kills with the Degreaser, Powerjack, or my secondary weapon which isn't part of the item set, all generated calling cards, for every kill. So it looks like whenever you kill or appear to have killed someone while you have the item set in your loadout, a calling card is generated. There is no catch.