Ancient treasures should remain with the finder. Although it is tempting to claim it for the government, fairness dictates it go with the person that put in the time, money, and effort. There can be exceptions, but these should be spelled out in advance, so both parties have a clear understanding from the start.
Searching for and discovering ancient treasures often takes a lot of knowledge, time, and expense. People who make that investment in discovering such treasures should have the option of keeping the treasures themselves or returning them to the governments that own the land or were the original owners of the property.
I believe it is only fair that ancient treasures remain with those who discovered them. It makes perfect sense, considering that a particular individual or group put the time and effort in to uncover the item. After all, just because a particular item is found in a certain location, it does not mean that it is automatically the property of that country. There are various forms of agreement that can be reached in these instances, all of which I believe should go in favor of the actual person or group who discovered the item.
I believe that if an ancient artifact if found, it should be purchased from the finder by a museum or other appropriate service for a set amount of money so that it may be preserved. Credit should be given to the finder by means of a plaque or something near the display of the artifact. While finding an artifact is a special thing, it is more special to allow the world to enjoy it as well.