That would (possibly) occur on partial days, but not when the sun is opposite your location, as light (no matter how near) cannot travel through the earth. Perhaps a good portion of the area around the "dark side" will be a dusk setting, but not the entire earth.
Edit: On partial days, the portion would still be in the same, elevated position, so it'd just put out a less strong light. Like a cloud in front of the sun.
And if we're really assuming it outputs the same light as the sun (which he didn't mention in the OP) then our eyes -- if they even came to exist -- would have evolved totally differently, as it'd be near-impossible to see with that much light. I'm leaning toward some sweet sonar action.