We live on opposite sides of the Walmart. The second one is fine

"We live on the opposite sides of the Walmart" makes me think that you live in the produce section and he lives in the bakery (or perhaps gardening supplies in a SuperWalmart). I don't think that's what you mean. I would try "He (or she or they) lives on the other side of Walmart."

We live at the opposite side of the Walmart.