Each state DMV's procedures are different. I'd start with Ohio's BMV (Bureau of Motor Vehicles') website:
http://www.bmv.ohio.gov/. There you can find contact info to find out an answer to your question.
I live in NY and I believe the last time I saw something like this type of transaction, they had a form to fill out when registering the vehicle that asked for the vehicle's info, seller, etc, and then asked for the value,( I think this mainly applies to newer vehicles over 7 to 10 years old). It was up to the DMV to then figure out if any sales tax was due.The form that was used dealt with transactions involving gifts, trades, inherited vehicles, etc. and came from the state tax dept. for the DMV to use. Perhaps Ohio uses a similar approach. But contacting your state's motor vehicle office or a local branch of it should give you the info you need.