When Artifacts are Discovered, How is their "Loot" Handled?

The answer is going to depend entirely on the antiquities laws of the country where the objects are found.

For example, in Mexico, generally any archaeological findings (as defined by federal law) are considered part of the national patrimony and so belong to the federal government, regardless of private or public ownership of the land where they were found.

In contrast, in the USA, any artifacts (human remains and sacred Native American objects exempted) found on private property are the property of the land owner, while any found on state lands (depending on the state) mostly are stored in a state repository. Likewise, federal agencies that manage land (such as Park Service or BLM) are responsible for the archaeological materials on their property and generally coordinate with the agencies within whatever state they own property to store any artifacts recovered. That said, there are limitations related to development that restrict developers from just plowing through archaeological sites in order to raise a building, even if they own the property.

In general, there is a spectrum of national laws between Mexico (and Israel) which have some of the strictest antiquities laws and countries with much more lax laws (or no antiquities laws at all, though these countries are becoming increasingly rare).

More generally, these sorts of situations where there is an interest in using the artifacts/land for some use other than pure research, one option is to conduct some preliminary research on the archaeological site/the artifacts that are in danger so as to "mitigate" the damage potentially cause by say, melting down gold or raising a building. The ideal would be to preserve the artifacts/site in perpetuity, but this sort of compromise were archaeologists get a set amount of time to study the objecst/site before they are destroyed is fairly frequent when you have multiple interests involved beyond just archaeology.