Player A Uses Exchange with a hand full of Dark World Monsters
Player B Takes Gold
During Player B's turn they use Graceful Charity and discard the same Gold.

Would Gold's second effect be activated?

It doesnt matter who's Hand Goldd is in. It will only get its Second Effect if it is Discarded by its Opponents Effect.

So No to your Question because Goldd now has a new owner and it will be played as if it belonged to that Person. So Graceful will only activate its First Effect.

Understand or Do you Need a Better Explaination.

Already understood the reasoning behind it. Just wanted a clarification.