There is hereby authorized and established a program of litigating civil trials, including domestic relations matters, whereby the litigants, by agreement, may retain the services of any retired justice of the supreme court or superior court or retired judge of the family court or district court to hear the merits of the issues before the court, all without a jury. Such trials shall be conducted in private, the cost of the proceedings shall be borne by the parties, and any judgments issued thereunder shall have the same effect as judgments of a court of competent jurisdiction and may be appealed to the supreme court.