Unclear whether you want phrases corrected, more phrases, or whole sentences. In my side, - seriously strange, as it suggests something physically internal. Here, the wound is in my side. on your side, - ok - fear not, we are on your side in this. from our side, - ok - from our side of the fence it looks bad. From our side, the arguement looks thin and has little benefit to us. (more liekly to use From our perspective or similar) about their side, - more than a little strange - The holsters hung at their sides as if ... I will tell you about their side of the dispute. ... by his side ...fine - She was by his side through the ordeal and ... then ran off. I'm afraid that you must take sides: There is no way to sit on the fence in this. Side with him and you risk losing all you have fought for. Choose sides, now. After this you will live with you decision. He was from the wrong side of the tracks. She was from uptown. This cannot be resolved by either side alone. Each side must be seen to compromise to make the negotiation work. It is a game of two sides. The two teams compete for .... The side that ... wins. Their side lost. Those on his side were much better at ... "side" in this way polarises to adversarial, that is, it focuses on the distinction between sides. Many times, you will not want "sides" but to stress cooperation, hence another word may be better. stakeholders. those involved. Enjoy. Hope you find something useful in all this.

Thank you so much for your answer, more than useful! I really enjoyed reading all these examples. Very helpful.