I am unsure if they are called the 5th and 6th officials, the goal line officials or something else but I am referring to the uniformed officials that stand on the goal line during the UEFA Champions League matches. Why are they positioned on the same side of the goal as the assistant referee for that half of the field - wouldn't it make more sense to place them on the opposite side of the goal to cover even more of the field?
The laws of the game clearly delineate where the referees should position themselves. The additional assistant referees (the official name for the goal line refs) are supposed to be on the same side as the assistant refs (official name for the sideline refs) because the referee positions himself at the top corner of the box on the opposite side of the field from the assistant ref. as such, he often cannot see the portion of the penalty box closer to the assistant ref and when there are lots of players in the box, the assistant ref also cannot see since he is 20-30 yards away. By placing the additional assistant referee there, the referee can see the box from the front and left side (when looking towards the goal) and the additional assistant referee can see the back and right side.
Here's a link to the rules of the game, where the additional assistant referee positioning is shown. This may help illustrate my point.
I can not comment as a new user. In addition to the answer provided by @Duncan being on the same side as the "regular" assistant referee makes it possible for the referee to see them from his/her regular position. Remember that the position on the field for the referee should "follow the diagonal". Should the additional assistant be on the opposite side, they would be in almost the same spot as te referee and/or outside of the direct line of vision. In order to judge if the ball has crossed the goal line you have to be on that line, the referee would rarely be in that exact position.