There is no rule. In most local transactions the seller pays, either by installing a system before closing or by issuing a credit. Whoever pays, get the work quoted before the inspection response deadline.
More detail
Radon is a negotiated repair item like any other, and neither Ohio law assigns the cost to a party. What actually decides it is the contract, the inspection response deadline, and how much competition the house has. Three ways it usually settles in the Greater Columbus and Central Ohio market. The seller installs a system before closing, which is the cleanest outcome for a buyer because the work is done, the post-mitigation test is in the file, and the warranty starts fresh. The seller issues a closing credit, common when the timeline is tight, though a credit sized off a single quote often lands low if the house turns out to need two suction points. Or the buyer absorbs it, which happens in competitive situations and is a reasonable trade when the home is otherwise right, since a typical system runs $1,200 to $2,400 here. The practical mistake is negotiating the number before anyone has scoped the job. Foundation type moves the price more than anything else: a slab-on-grade house with an unobstructed exterior wall is at the bottom of that range, and a crawl space needing a sealed membrane is a different pricing tier entirely, $1,800 to $3,500. A ten-minute phone conversation about foundation type produces a range accurate enough to negotiate against, and it costs nothing. If the response deadline is days away, ask for the quote first and the negotiation second. A written scope beats a round number in every conversation that follows.