It does not say, “I beseech Euodias and Syntyche,” but the Apostle (the Spirit through him) beseeches each sister personally. It is a personal appeal to you and a personal appeal to me to reconsider our views and our attitude of heart towards one with whom we may be having difficulty.
It does not say, “I beseech Euodias to be of the same mind as Syntyche,” or, “I beseech Syntyche to be of the same mind as Euodias” either. Rather, “Be of the same mind in the Lord.” This is truly searching for each. Why would we not be of the same mind in the Lord? There must be an obstacle with one or the other personally! We may have differing views about some Scriptures or some matter, but we can still be of the same mind about them. That is, we can leave the other room for exercise and liberty of soul in many things in which we differ.