Whom the equipment piece is for is more complicated, but it appears that whoever strikes the killing blow has ~40% chance of receiving the drop, with the remainder divided equally between the other characters. If a drop is given, it has a 50/50 chance of being a weapon or an armor. The drop chance is unaffected by Overkill, who kills it, or any other factor. Some enemies always drop equipment, whereas others drop some only rarely. Unlike the item drop chances, which are always the same, the equipment drop chances vary between enemies.
Not every opponent drops equipment, although quite a few do. Opponents can drop weapons and armor pieces, but the equipment drops are separate from the item drops. Most equipment pieces come with auto-abilities, but once the player reaches Guadosalam, they can also customize auto-abilities onto equipment as long as they have the required number of items, and as long as the equipment has at least one empty slot. Pieces of equipment can have a maximum of four auto-abilities. See also: Final Fantasy X auto-abilities, or Equipment drop mechanics