In the UK you have to declare the item, and local museums are then allowed to bid a fair market price for it, which you have to accept. This money would then be split equally between the person who discovered the item and the landowner. It works pretty well - there's an incentive to legally declare what you've found, as you'll get paid for it, and by doing so the archaeology is often preserved and the item doesn't just disappear into a private collection.

Ref: https://en.wikipedia.org/wiki/Treasure_Act_1996