  We have studied the dielectric responses of the layered cobalt oxysulfide
Sr$_2$Cu$_2$CoO$_2$S$_2$ with the CoO$_2$ square-planes. With decreasing
temperature below the N\'eel temperature, the resistivity increases like a
semiconductor, and the thermopower decreases like a metal. The dielectric
constant is highly dependent on temperature, and the dielectric relaxation is
systematically changed with temperature, which is strongly correlated to the
magnetic states. These behaviors suggest that carriers distributed
homogeneously in the paramagnetic state at high temperatures are expelled from
the antiferromagnetically ordered spin domain below the N\'eel temperature.
