Hi, I'm right handed and I injured my R shoulder 8yrs ago. I have had 3 surgeries on the shoulder and it took about 6yrs for a doctor to get it right. In the meantime, I was using mostly my L arm for almost anything since my R shoulder/arm was in so much pain. As a result, I ended up having L shoulder surgery
5 months ago. It's not that your pain goes to the other shoulder, but it's caused by over compensation for using your other arm more to avoid pain on your hurt shoulder. It does happen and especially if the arm/shoulder you hurt originally is your "main" one. Good luck!