I assume the intention of this is that it can only discard your own wounds, not other investigators, although it would make sense that you could use this on others. Is there some rule preventing you from using this on another investigator? Or preventing you from using it on another investigator... on the other side of the map?

Bandages: " Action: Discard up to 2 facedown damage. Then discard this card."

Yeah, you're exactly right. As an action (your action), you discard the facedown damage from yourself. It's always 'you' if not specified otherwise. Whilst it seems logical in the real world to be able to apply it to someone else (especially in the same space), it just doesn't do that.

We had a game where we were wondering if the Trade action was even worth picking the bandages. Later in the game, it would have become very useful for the Investigator that actually didn't pick them up. I just love how this game has pros and cons to each decision.

The wording of the card indicates that the person activating these an as action discards Damage. So that would be you, not another investigator. That said, it would be completely reasonable to house rule that you can use these on someone else in the same space, as it fits the theme of what they are and can do. But definitely not across the map!