Lets say there are two DS players going against the LS. Lets say one DS player runs out of cards. What happens at this point....does that particular player go out of play?

The challenge doesn't override DS auto-loss conditions, so a DS player with no cards in their deck (be it their objective or draw deck) is eliminated, and the other DS players continue without them.

Cool, that is the way we played it. Thanks, PBrennan.

The win condition for the LS challenge deck is to eliminate both players Command deck. As soon as one of those players is eliminated, the cards they control are removed, leaving the remaining players.