There is no gun registration in Ohio, but firearms could be tracked from point of sale when the firearm is new by the information stored in the dealer's bound book. After that, there is no law requiring a paper trail and it's up to the individual to keep track of these things. Lot's of people don't.
A simple bill of sale is a good idea to protect both the buyer and seller. Most of these documents have a statement that says the buyer is not a felon and is legally allowed to own the firearm. When the buyer signs it, that provides some protection to the seller since the seller cannot knowingly sell to someone who is not allowed to own the firearm. Both the buyer and seller should check IDs to verify that the person matches the name on the bill of sale. If the buyer purchases a firearm that was used in a crime and the authorites come looking for it, it's good to have a document that says "I purchased the firearm from Joe Blow."