Vicente Martinez, 18, was apprehensive the first time he visited the former drug house in Syracuse, N.Y. The building had been transformed into the Center for Court Innovation’s peacemaking program, where issues can bubble up and be resolved outside of the traditional criminal justice system. “I was messing up a lot,” Mr. Martinez recalled of his two previous stints in jail for driving a stolen vehicle across county lines. “I was doing dumb kid’s stuff.”