I have dreams where I feel pain, similar to yours. I've had it a lot when I was young, but nowadays, it's rare. For me, the pain's not in my side, it's in my back. Exactly the middle of it. It's not exactly right to say I feel pain, but it's definitely uncomfortable and disturbing. It's sort of like repeated thumping to the middle of my back. Sometimes it's brief, sometimes longer. After I wake up, the "pain" is completely gone.
For example, I had a dream where I was standing on a roof, and a giant bee landed on my back and stung it. But instead of the actual bee sting pain, it was that same repeated thumping behavior on my back, then I wake up.