The following website defines "共同利用型病院" as "中核となる救急指定病院に当番で他の病院や開業している医師が集まり、休日や夜間の救急医療に当たる方式" (A central hospital designated as an emergency hospital where medical practitioners and physicians at other hospitals take turns to go on duty to provide emergency care on holidays and after regular hours): http://tatujin.net/kyuukyuuiryou.html.

The United States does not have a comparable system, so American English doesn't have an exact term for it. I would go with a literal translation, simply calling it a "joint-use hospital" and if permissible add a short description (e.g. joint-use hospital, a hospital jointly used by physicians from other hospitals to provide emergency medical care).