Ohio Rev. Code § 1901.37: Bonds.

All official bonds required by sections 1901.01 to 1901.37 of the Revised Code, when properly executed and approved, shall be filed with the city treasurer, except that in a county-operated municipal court, they shall be filed with the treasurer of the county in which the court is located. Whenever a surety bond is given, the premium on the bond shall be paid out of the city treasury, except that in a county-operated municipal court, the premium on the bond shall be paid out of the treasury of the county in which the court is located.