The geometry of viewing Earth in the Martian sky is the same as viewing Venus in our own Terran sky. The interior planet periodically approaches a certain maximum angular separation from the Sun. Just as we see Venus but not the Sun sometimes in the early morning or evening twilight, Martians (and rovers on Mars) could see Earth but not the Sun at corresponding intervals of Mars' rotation. It's a significant amount of time, up to several hours per day at maximum separation.

The angular separation between the Sun and the Earth as seen from a space probe is significant as far out as Cassini at Saturn. The probe can receive commands from Earth without the signal being overwhelmed by solar radiation, except for a few days each (Earth) year when Earth is too close to the Sun as seen from the spacecraft. (Earth doesn't literally go behind the sun often, thanks to inclination of the planetary orbits plus Cassini's own inclined orbit at Saturn.)

> Our line of sight to Mars is independent from Mars seeing the sun.

Our line of sight to Mars as an entire body is independent, but our line of sight to a particular point on Mars is indeed correlated with that point facing the Sun.