这句句子中的第二个with后面应当有一个解词宾语,由于原句为被动语态,其宾语是句子的主语了"the practice of ...".不妨还原原句（需要加一个主语）："As we shall see,we are experimenting with the practice of housing the psychologically disabled with the willing families in the United States today." 这样,对你理解有所帮助.
...experiment with ...是指：“用XX进行试验/实验”“对XX进行试验/实验”.