It is possilbe and legal for you to get 1099's from prior years service, in which they were previously neglected. The person or business issuing them, would also have to provide a copy to the IRS, who will fine them for late submission.
The fine is non-punitive and merely covers the cost of filing the 1099's out of time: 50 to 75 dollars each occurance.
You would have to file amended tax returns for those years, and pay any taxes associated with 1099 income.
in so doing, you would be able to take business related expeses if any and would use either schedule C or C-EZ:
YOu would have to pay SS and MC tax in the form of Self Employment taxes of 15.3% of your net earnings as determined on the schedule C, using the following form to figure the SE tax:
this will cause changes to your 1040 for the year in question, and you will have to use 1040X, if you already filed your taxes for those years.
this will of course cause you some extra expense, and you may have cause of action to recover any additioanl expenses cause by the un-timely filing of the 1099's in small claims court. (depending on the amount of the additioanl expenses).
Note additional expenses are not the taxes you have to pay, but any penalties, interest, and tax preparation fees.