Pilfered artifacts are a problem for archeologists, but the problem can go much deeper than just the loss of the items.
After reading this post on the problems of the fossil black market, commenter and archeologist trudibell_ chimed in to tell us that the pain was shared across archeological digs as well — and not just because of the loss of the artifacts. It's also losing the chance to look at those artifacts in the right context:
Looting is a serious problem for us too, obviously. Professional "pot hunters" or the like ruin stratigraphic contexts making it impossible to reconstruct the past. Simply put, underneath our feet the earth provides us with a timeline of events, those that are buried deeper occurred earlier while those shallower occurred much more recently. When a lot of hobbyists grab their shovels, very rarely do they record all of the nuances in the soil and other artifacts that are uncovered during your standard looting. In this way, there's nothing science can actually say about an object other than "neat."
This is one of the main reasons why supporting any sort of private-profit motivated digging process is so dangerous, as you put it, fossils are a precious, limited resource and without proper excavation once they are removed from the ground there is no way to attempt to reconstruct the surrounding environment, rendering the object nothing more than a potentially expensive curiosity and we learn nothing.