Ohio Revised Code Title XXV. Courts Appellate § 2503.41

Process issuing out of the supreme court shall be directed to the sheriff or other proper officer of the county where it is to be executed, who shall serve and return it. The marshal or a messenger of the court may serve a rule, order, warrant, or notice of citation issuing out of court, for which he shall be paid the same fees as other officers are entitled to for like services.