I think "on the one side" is even more rare than the Ngram indicates. Leafing thru the hits, many examples don't fit this context (e.g., "six cubits wide on the one side," or, "the join fields on the one side of a one-to-many relationship," or, "every advantage gained on the one side would be a disadvantage on the other.").
In my ignorance (or innocence) I'd always assumed The grass is always greener on the other side was a direct reference to the Three Billy Goats Gruff. Re the proverb occurring in Japanese, this is very possibly a borrowing: Japan has fully embraced quite a few proverbs from the Western canon, even when they already had perfectly good homegrown equivalents, and the form is so similar here that ...
Other hand is just right, even in that position. Stonehenge is being compared to some other site. Side is used when comparing two aspects of one item: On the one side you have her stiff acting, and on the other, you have the breathtaking beauty.of Marilyn Monroe to stare at.
on the other side vs on the other hand - English Language & Usage Stack ...
I've come across the formulation by one side, by other side instead of on the one hand/side and on the other hand/side. I strongly suspect this to be wrong and maybe Brazilian Portuguese originated...
Is "By one side, ..., by the other side" a correct expression?
"On the other side" suggests that you and the other party are in two different territories, separated by a border or barrier. "On the other end" connotes a real or virtual line or conduit that stretches between you and the other party, such as a road or a network connection.