We study light propagation in a neutral Weyl semimetal with the Fermi level lying at the Weyl nodes in the weak self-interacting regime. The nontrivial topology induces a screening effect in one of the two transverse gauge fields, for which we find two branches of attenuated collective excitations. In addition to the known topologically gapped photon mode, a different massless and slightly damped excitation appears. Strikingly, at low energies, this excitation has a linear dispersion and it propagates with the same velocity as the electrons, while at energies well above the electron-hole continuum threshold it behaves as a massive attenuated photon with velocity similar to the speed of light in the material. There is a crossover at a certain momentum in the direction perpendicular to the separation of the Weyl nodes above which this gapless mode enters into an overdamped regime. Regarding the unscreened gauge field, we show that it is also attenuated, which is a nontopological property shared by Dirac semimetals as well.