Live in Illinois work in Missouri. Do I need to send copy of states return to the other state?

There is no reciprocity between the 2 states.  Since you work in Missouri you'll need to file a non-resident income tax return. Do this one first. Illinois will also tax the compensation but give you full or partial credit for the Missouri income taxes. Turbotax will take care of including the required forms when you e-file. 

No. The program will send state returns that are required. This doesn't mean you may not get a letter from one of the states asking for a copy of the other state. Neither states it as required.

MO FAQ states: You must allocate your Missouri source income on Form MO-NRI and complete Form MO-1040. You must include a copy of your federal return with your state return.

IL instructions state: Include any required attachments (e.g., Schedule M, Schedule ICR, Schedule IL-E/EITC, Schedule CR, federal Form 1040 and Schedules) with your Form IL-1040.