(1)    Any 2 or more counties may jointly, by majority vote of all the members of each county board, provide for a county home, infirmary, hospital, or similar institution, or juvenile detention home, which shall be established, maintained and operated pursuant to all the statutes relating to the establishment, maintenance and operation of similar institutions, respectively, by any single county whose population is less than 250,000, except as otherwise provided in this section; and in all respects, except as herein specified, each such institution shall be the county institution of each of the counties so joining.

   (1m)   Any 2 or more counties may jointly, by majority vote of all the members of each county board, provide for a secured residential care center for children and youth, as defined in s. 938.02 (15g), under ss. 59.52 (7) and 66.0301. A secured residential care center for children and youth established under this section shall be the county secured residential care center for children and youth of each of the counties so joining.

   (2)   The county board of each such county shall elect 3 of its members to represent the county in a joint committee to select the site, anywhere within either of said counties, and prepare the plans and specifications of the buildings, for such institution; and said joint committee shall be vested with all functions, relative to the establishment of such institution, possessed by the county board relative to like county institutions.

   (3)   Upon approval of the site, plans, and specifications for the institution, as provided in ss. 46.17 and 301.37, the joint committee shall report to the several county boards the estimated cost of the site and buildings, and the amount thereof chargeable to each county on the basis set forth in sub. (6) (a), appending to each report a copy of the plans and specifications and all matter relating to the site and buildings. If the report is approved by each county board, the joint committee shall purchase the site and cause the buildings to be erected in accordance with the plans and specifications.

   (4)   The functions of the joint committee shall terminate upon completion of said buildings; and the board of trustees thereupon elected shall be composed of 2 electors from each county when 2 counties join, and of one elector from each county in all other cases; which trustees, except those elected at first elections, shall hold during terms of 4 years in the former case, and as many years as there are counties joining in the other cases. Trustees elected at first elections shall hold for terms beginning at once and so ending that the terms of one member only will expire on the first Monday of each succeeding January, beginning with January of the second year thereafter; which terms shall be allotted to the respective trustees at their first meeting. The county board of each county joining shall at an annual meeting elect a successor to any trustee from such county whose term will expire on the first Monday of the next following January. The trustees shall have the qualifications and functions and be subject to the limitations and restrictions of trustees elected pursuant to s. 46.18. They shall be reimbursed their traveling expenses necessarily incurred in the discharge of their functions, and shall receive compensation similar to that of the trustees, respectively, of like county institutions; and if the county board is empowered to fix the latter, the compensation of the joint trustees shall be fixed by joint action of the county boards.

   (5)   All receipts on account of such institution shall be paid into the treasury of the county in which said institution is located.

   (6)   Prior to the election of the trustees the joint committee, and thereafter the board of trustees, shall audit all claims against said counties on account of such institution; and all such claims shall, in the first instance, be certified to and paid by the county in which said institution is located, subject to apportionment and adjustment among the several counties as follows:

      (a)    All expenditures for the site, buildings, furniture, equipment, and permanent improvements, after deducting all receipts therefor except county appropriations, shall immediately upon payment be apportioned by the clerk of the county in which the institution is located and certified to the clerks of the other counties, on the basis of the percentage which the valuation of the taxable property in each county bears to the valuation of the aggregate taxable property in all said counties, as determined pursuant to s. 70.57.

      (b)    All expenditures for repairs, maintenance, and operation, after deducting all receipts therefor except county appropriations, shall be so apportioned for each month, on the first day of the next succeeding month, on the basis of the percentage which the aggregate cost of keeping the inmates at public charge from each such county bears to the aggregate cost of keeping the inmates at public charge from all such counties, adopting as the unit of cost the total average cost per capita per week of keeping all the inmates, at public charge and otherwise, in said institution.

      (c)    Immediately upon receipt of such certified apportionment each such other county shall pay over to the county in which said institution is located its proportion of said expense.

   (7)   

      (a)    If any net profit shall arise from the operation of said institution it shall be apportioned among the several counties on the basis prescribed in sub. (6) (b); and the county in which said institution is located shall pay over to the other counties, respectively, their proportions of such profit.

      (b)    The board of trustees shall maintain as a segregated cash reserve the 2 percent charge required under s. 46.18 (13). It may from time to time appropriate from such reserve sums to be expended solely for the enlargement, modernization or replacement of such infirmary and its equipment.

   (9)   At any time after the organization of any such institution, any additional county or counties may join in the support and conduct thereof upon payment of such equitable proportion of the original cost of its establishment, and any joint county may withdraw upon such terms, as may be agreed upon among the county boards of the counties interested; and thereupon the board of trustees of such institution shall be reorganized, in such manner as may be determined by the county boards of the participating counties, to conform to sub. (4).