The protonic conductivity in the hydrogen-bonded chains is investigated theoretically in the framework of the two-stage transport model. The strong interactions with optical phonon stretching mode are considered. We obtain the phonon-induced transition from the insulator to the metal-type behavior from the temperature dependencies of the hopping conductivity and analyze the influence of the specific Grotthuss mechanism on the transition character. We investigate also the main peculiarities in the frequency dependencies of the band and hopping conductivity parts which appear due to above-mentioned two-stage mechanism of the proton migration along the chain.