British law actually provides for something like eminent domain when it comes to “treasure trove”: The finder (probably*) doesn’t get to keep the actual ancient gold coins, but they get paid fair market value in modern money by some museum.
*Only if no museum is interested would the finder of treasure in the UK just get to keep the actual physical treasure.
I think the British approach is the best of both worlds, really. Apart from maybe rolling around in them like Scrooge McDuck, I’m not really sure what I’d do with a hoard of ancient gold coins anyway. Get paid modern money I can use; and have a nice little sign in the British Museum with my name on it, telling future generations the story of how, after I found the ancient treasure, I did the right thing and notified the proper authorities (instead of trying to sell priceless antiquities on eBay). Win-win!
I don’t know what the Italians, do, though; or the exact circumstances of that find. Maybe one way or another no one gets paid; or alternatively, maybe those coins get auctioned off to private bidders to wind up in some rich guy’s collection.