Because of a problem with really painful tendonitis in my left shoulder, I have switched from using the mouse in my left hand to using it in my right hand. At first, this felt quite awkward, and there were times when I felt as though I just couldn't do what I wanted to do. Even so, the process turned out to be easier than I expected. After a few weeks I am using the mouse full time in my right hand, with only slightly less precision than I had when I used it in the left. I expect that it won't be long before I am just as good with the right as with the left.
This doesn't mean that I think switching handedness is easy! Not at all. But many right-handed people use the mouse with their left hand so that they can do other, more skilled tasks (such as writing) with the right. And many lefties use the mouse in the right hand for the same reason. Using a computer mouse doesn't really require the same level of skill, strength, or dominance that is called for in certain other jobs, especially handwriting.