The governments in question have laws stating that anything found in their soil belongs to the government. These coins are being returned under an international treaty dealing with "Cultural Heritage". The US is a signer of this treaty. basically what happens is a signatory country can petition to add a list of "cultural heritage items" that either originated in their country or could have been found in their country. Unless the owner of these items can show proof that they were exported with proper licenses from the country, or they were removed before a specific date they belong to the home country and the signatory country in which they are discovered is obligated to return them.
Most of the countries list antiquties and art object etc, but several countries have added coins to the list. Greece and Italy clain all coins before about 1500, same for Cyprus. China has been trying to add all coins before 1911, At one point during the first gulf war they were considering adding anything from Iraq from before 1990! I'm sure other countries will be trying to add their coinages as well.
This is a real headache for collectors of ancient coins. For example there are tens or hundreds of thousands of roman coins that were produced, found all over Europe and they have been bought and sold among collectors for hundreds of years. Most of these coins are of low value and nave never been plated in sales, and most of them do not have extensive pedigree listing going back decades or hundreds of years because they weren't that special. So the average roman coin collector can not provide written documentation for each coin in their collection proving it was removed before say 1950, or show an export license for each coin. All of those coins are now liable for seizure and forfeiture back to Italy.
Now in most cases that isn't going to be a problem, unless they take them out of the country and then try to bring them back in. If they do that the customs agents are supposed to confiscate them. It can also be a problem if the collector tries to buy a coin for their collection from a dealer outside the US. Since most of them don't have the documentation they can be confiscated coming into the country. (That would also give them justification to examine the rest of their collection for "proper documentation" as well.)
The interesting thing about the case in the article is he wasn't convicted of importing contraband coins, he was convicted of ATTEMPTING to even though the "coins" he got were modern fakes. But as part of the penalty they took twenty coins from him that he WASN"T charged with importing illegally. (Apparently they were already in his collection. I guess he didn't have documentation on them.)
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Did the surgeon even know that they were 'illegally taken'?
From earlier discussions on this, apparently he did know they were supposedly illegally exported form Greece. So he did believe he was dealing with "illegal" coins.