The Earth is opaque - as you can verify if you look at any rock.
There would be a vanishingly small amount of light scattered through the atmosphere, but not so that it could be either measured or noticed.
(a good exercise for the student! calculate the photon flux at the antisolar point from scattering through the atmosphere, assuming Rayleigh scattering)
The question is legitimate. In fact, after sunset, we see well despite direct sun light is no more visible. Our eyes are no longer reached by direct sunlight, we perceive a good illumination from the light diffused by the atmosphere. The relevant Rayleigh scattering equation has an angular component (1 + cos ^ 2 (theta)), which is maximum for small angles forward and backward. After sunset, the angle theta gradually increases, and this angular dependence, together with the greater distance between the observer and the air molecules that scatter sunlight, explain why the diffused light gradually decreases and the darkness takes over.