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, can you please confirm? (I only know the author comes from Rio, I don't speak Brazilian Portuguese).
Those are not grammatical in English. They are inappropriate word-for-word translations of the common Portuguese phrases por um lado and por outro lado.
In idiomatic English they should respectively be on one hand and on the other hand.
The principal error derives from the Iberian habit of omitting determiners in front of their words for “other”, and the translator not realizing that. The secondary error is literally translating side where in English we use hand in the corresponding constructions.