"The friend is on their right side."
You are saying their friend is on the right... On the right of what?
Your sentences would be translated as: Their friends is on the right side. 'Their' in the question (The friend is on their right side) is refer to 'right side', not 'friends'. So it would be 朋友在他们的右边
That sentence means their friend is on their side. It shouldn't specify who's friend it is.