PROVIDENCE, R.I.—A retired police officer and the proprietor of an organic eatery make an odd couple when it comes to trying to overturn marijuana laws in this tiny state, but Jack Cole and Josh Miller are giving it their best shot.

Mr. Cole, 71 years old, is a veteran of decades with the New Jersey State Police, almost all with the drug squad. Mr. Miller, 55, runs Local 121, a restaurant favored among "buy local" diners, and also serves in the state Senate, where he leads a special commission to study marijuana prohibitions....