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If these are 1,000 year old coins found in an archeological site.... who do the coins "belong" to that were "robbed"?

Depending on local law - the owner of the land, or the national antiquities authority, or the owner of the mineral rights, or...

"Finders Keepers" is very, very seldom the law, when there is anyone with a plausible prior connection to the found item(s).

Treasure trove laws, as applicable in the UK and US (and previously in Roman law), are laid out here:

Only the UK coincides with the largest extent of the Roman Empire. So these laws would be inapplicable to the vast majority of Roman finds.

Before the archeologists had a chance to rob them first.

Robbing implies taking property unlawfully by force or threat of force. Who exactly was threatened by force in archeological sites?