My first experience with the left/right divide was a one-day course "Drawing on the Right Side of the Brain" at SIGCHI 98(?). Based on her book and a longer course, it was the first time Betty Edwards had tried to condense her originally multi-week course to a single day. It worked for me.
During the course she referred repeatedly to L-brain and R-brain as metaphors because at the time medicine and psychology were busy denigrating any reality of left/right brain differences and she had clearly been beaten up by the "experts". I haven't talked to her since then, but I'm sure the last decade's real research on the differences has gratified her.
There were two exercises that made an impression. I hadn't read the book before the course, and came into the class doubtful that I could learn to draw. The exercises had a profound impact on me. I was able to draw, and I felt the mental shifts she described as necessary to do it. I will briefly describe them here.
The first exercise was to copy Picasso's "Portrait of Igor Stravinsky". I have tried to copy other images in my life, so I was quite aware of my inadequacies in doing so. The second exercise was to draw my hand, and I also doubted that I could do that. But I was open to learning.
The trick is to copy Picasso's "Portrait" upside down so as to disengage the left brain. I started in, and was really happy with how well I was doing, both in speed and accuracy. Until I got to the hands. Without thinking about it, I watched my drawing turn to crap. I erased the crap and forced myself to look only at the individual lines. Drawing each line segment very slowly, I faithfully reproduced the hands. After that I relaxed into my previous speed and accuracy until I got to the head. I was ready that time and went into the slow, one-line-at-a-time mode and finished the head. Of course, this is all described in the book, which I looked at later.
The exercise allowed me to experience the switch between left and right hemispheres. Surprisingly, there wasn't any feeling of clumsiness in my drawing hand when the change occurred. Apparently the brain is well practiced at making the transition. One lesson was how subtle and natural the change is. A second lesson was how easy it is to induce the change once you know it can be done. The third lesson was, of course, what a dramatic difference it makes.
During the second exercise (drawing my hand), I played with the change and became comfortable with it. I became confident that I could do it in the future as need arises. I don't often draw, but the lessons I learned have benefited my work every since.
The lessons also opened me to thinking about left/right brain issues. The academic condescension that was rampant at the time in most cognitive literature kept me from reading more about it until recently. Now it seems that a corner has been turned, so I am immersing myself in the subject.
The overall lesson from the experience was that drawing is mostly seeing. Betty made the point that drawing isn't art, so learning to draw didn't mean we had become artists. She sees drawing as a basic skill that everyone should learn, like doing arithmetic or using a keyboard. This message is more subtle than you might think since many of our personal and social problems arise from not seeing clearly.
A corollary to the overall lesson is that most of us can really only draw well an actual (physical) object. Asked to draw a house, we would best draw an actual instance of a house. Asked to draw a person, we would best draw a real person. Drawing a generic house or person will usually lead to an uninteresting result. I don't know how artists draw things that are entirely in their heads.