Every county BOE has a data base of all registered voters. If someone requests and is sent an absentee ballot, they are flagged in the appropriate poll book and must vote provisionally. The check to see if they already voted is done during the counting.
Ohio now uses electronic poll books, but it was fairly simple even in the paper book days. The poll books are printed just before election day and there is a supplemental sheet sent out before the polls open, so each precinct knows who has already voted.