We consider two closely spaced slits of subwavelength width, pierced in a metal screen, that are illuminated by short, arbitrarily polarized, plane-wave pulses. The $p$-polarized component generates plasmonic pulses propagating on the metal surface between the slits. In addition to the primary transmitted pulses, scattering of the plasmonic pulses by the opposite slit is found to produce secondary (echo) radiation pulses, which causes splitting, shaping, and time-dependent polarization modulation of the radiated pulses in the far field. We study these effects in the linear regime. A previously explored phenomenological model is extended to the time domain and is validated by rigorous numerical calculations. Our analytical and numerical results have implications for the transmission of pulsed light through arrangements of nanoscale apertures with plasmonic coupling.