My (native Korean) girlfriend said 이편, 그편 and 저편 are no longer in use. I asked her what they meant, she instantly replied, I don't know. She looked it up on Naver and it turns out that 이편 is exactly the same as 이쪽.
She also said she would use this word if she wanted to ask someone to send something 이편으로; and also, 이편이 좋겠다: this way (so, doing it this way) will be good.
The word "side" has probably very little to do with this.
이便, 그便, 저便, 건너便 (the other side of the road from an earlier lesson). If it means side, it's not as a location as we think of it, but more the side of an exchange in some adverbial sense: trip, flight, service, sending something by mail, food passing through you (it's part of the word for toilet, and means excrement, among other things, at least in Japanese), . . .