"But my question here is, what makes you think the 1099-G is going to be mailed to the taxpayer whose identity was stolen, anyway?"

In Illinois, there have been many thousands of fraudulent on-line applications for unemployment benefit.  The state is mailing follow-up notices to the original taxpayers at their state income tax filing addresses.  The taxpayers did not even know their identity had been stolen until they receive the follow-up notice.  So, yes, I expect they will receive 1099-G forms showing the unemployment benefits electronically paid the fraudsters.