Does anyone know if items that affect the drop rate have to be equipped on the character you're controlling to be effective? Or do these items work for anyone in the active party?

For instance if my party is Adol, Laxia, and Sahad, and I'm controlling Adol, but Laxia has the Bandit Bracelet (Obtained Items ⬆⬆) and the Gambler's Coin (Rare item drop ⬆), does it affect the drops, or does it only do so if I'm controlling Laxia?

Works the same way as any other equipment, it affects only the character that wear the item. If someone in your party has the Bandit Gloves/Bracelet/Gauntlets but you're not controlling that character you have the normal chance for drops. However if the AI kills the NPC with that character the better drop rate kicks in.

Yes, you have to control the character wearing that equipment to make your kills drop more loot. I own the PS4 version though but I guess this is the same on every port.