Should Ohio ever require background checks on private sales, (which I highly doubt in the near future) I would predict massive non-compliance, and it would essentially legislate the law abiding private seller into criminal status.
Of the hundreds of millions of guns in this country, the majority do not have any paperwork tying them to the current owners - mostly just those purchased new from dealers, and are still held by their original owner. Other than undercover sting operations, if a private party sells a 2nd hand gun to another individual, and does not want to pay for a dealer to do a background check, it would be totally unenforceable, and any fool who believes that black market street dealers, who are mostly already under disability are going have dealers do background checks and transfers, probably believes criminals respect gun free zones as well. Over half of my guns were purchased used, and if I sold them to my buddy, they would have to prove he purchased them after such a law was passed, which would be impossible unles sthey witnessed the exchange, unless, of course, they managed to pass gun registration first.
The support that "universal background checks" gets in polls is mostly from people who are ignorant of current laws, and what that actually means.