You are not required to issue a second copy to the contractor for the state return. The income is reported on the federal and the state returns but the Form 1099-MISC is not sent with the tax return like a W2.
Each state has it's own rules as to whether a copy is required to be filed with the state, like the federal rules, in relationship to the payer. If you aren't sure if you are required to file the forms with your state, contact our state's Department of Revenue.