The towns of Clinton and Madison may appoint, at annual or special town meetings, two or more special constables, who may inspect and measure shellfish and shells taken by any person in one day from the portion of the Hammonasset River over which such constables have supervision, authority and control as provided in § 26-278, and may arrest, without warrant, within such portion of said river or within the town to which the officer making the arrest belongs, any person found violating § 26-276, or who has in his possession shellfish or shells taken, gathered or collected by him within said limits in violation of said section. Said towns shall respectively provide for the payment of such special constables as are appointed.