The commission shall select a site for the building and may contract with the owner to acquire the site by gift or purchase, or the site may be acquired by proceedings in eminent domain in the name of either the city or the county, as the commission may determine. All land acquired as a site for the courthouse or city hall shall be owned jointly by the county and the city, in the same proportion as each contributes to the acquisition of land and the construction of the building. If the entire tract is acquired by either the city or county, it shall convey the proper undivided interest to the other.