Ohio Rev. Code § 5155.34
The boards of county commissioners of two or more counties may form themselves into a joint board of county commissioners for the purpose of establishing a "district home." For such purpose a joint board of county commissioners may use a county home site and buildings already established in one such county, or, it may purchase a new site and erect suitable buildings thereon, provided existing county homes in such counties, other than the one in which the established site is chosen, are to be closed and abandoned.
When an established site and buildings are used such joint board of county commissioners shall cause the value of the site and buildings to be properly appraised. The appraised value of an established site and buildings, or in case of the purchase of a new site and construction and of later betterments and additions, and the purchase of additional land, shall be apportioned among the counties of the district in proportion to the taxable property of each county as shown by their respective duplicates. After decision to establish a district home has been made the board of county commissioners of each county shall proceed to provide the necessary funds for the development and subsequent operations thereof.
If two counties create a district home the boards of county commissioners thereof shall constitute the board of trustees for the construction and subsequent management when ready for occupancy. If three or more counties join in establishing a district home, the board of trustees shall consist of one person appointed by the board of each county. The first appointment to such board of trustees shall be so arranged that the term of one trustee shall expire one year after the first Monday in February following the appointment, and one each year after such day. Subsequent appointments to the board of trustees shall be for the same number of years as there are counties in the district. Such trustees shall serve without compensation, but shall be entitled to receive necessary expenses entailed in the performance of their duties. Any appointed trustee may be removed by the joint board of county commissioners, for good and sufficient cause, after a hearing upon written charges. The board of trustees shall be responsible for the erection and subsequent management of the district home, when it is ready for occupancy. Such board of trustees shall appoint a superintendent or administrator and conduct the home in the manner prescribed by sections 5155.01 to 5155.33 of the Revised Code, insofar as such sections are applicable.
R.C. §5155.34