Do the coins that are on your Pirate Ship Mat need to disappear (go back to the coin pile) when you opt to use them in your buy? Or do they stay there for the entire game, being used whenever you have the Pirate Ship card come up and opt to do so?
You don't lose the Coin tokens on your Pirate Ship Mat when you use them in your buy.
I think the clue is in the fact that the rules refers to them as "Coin tokens", not just "Coins". Confusion between them and coins is understandable, but basically they're something completely different: the tokens only serve as a reminder of how many Coins you receive every time you play a Pirate Ship and choose its second option. Coin tokens do give you Coins, but they are not Coins themselves and are not expended in the same way as Coins!