To quote Ohio Revised Code 4301:01, “sale” and “sell” include exchange, barter, gift, offer for sale, sale, distribution and delivery of any kind, and the transfer of title or possession of beer and intoxicating liquor either by constructive or actual delivery by any means or devices whatever....
I suppose if they wanted to, Ohio's Liquor Control authorities could even enforce on beer that is gifted - or even shared at a homebrewers' club meeting, competition, fair, etc. since that would meet the definitions of "distribution" and "sale." Of course, I don't know whether that has ever happened, and I doubt it has. And I presume that, as long as one acts responsibly and doesn't get carried away, it is unlikely that the authorities would be alerted or take any enforcement action.
All that said, I've been known to give beer away, take it to club meetings, and enter competitions.