As far as I know, some contexts prefer one over the others. In the context of phone : The person on the other end (not at the other end or on the other side) -- I would use either on the other end/at the other end. Not on the other side.
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New2grammarI was on the other side of the road - meaning across
I was at the other end of the road - meaning the opposite end of the road.
Optilang, could you elaborate the difference? My understanding of across and opposite is the same.