Ohio Rev. Code § 2725.28
The fees of officers and witnesses shall be taxed by the judge, on return of the proceedings on a writ of habeas corpus, and collected as a part of the original costs in the case. When the prisoner is discharged, the costs shall be taxed to the state, and paid out of the county treasury, upon the warrant of the county auditor. No officer or person shall demand payment in advance for any fees to which he is entitled by virtue of the proceedings, when the writ is demanded or issued for the discharge from custody of a person confined under color of proceedings in a criminal case. When a person in custody by virtue or under color of proceedings in a civil case is discharged, costs shall be taxed against the party at whose instance he was so in custody. If he is remanded to custody, costs shall be taxed against him.
R.C. §2725.28