It is true that some brain functions occur in one or the other side of the brain, like language on the left side but there is no study that I know of that has shown a strong left or right-sided brain network. A couple of years ago a big study came out from the University of Utah where they scanned the brains of 1000 people while they did various activities, they then measured their functional lateralization (measures specific mental processes on each side of the brain). The team found patterns for left and right sidedness in certain connections but nothing to support that people are intrinsically left or right sided in their thinking. It is true that in extreme circumstances like cutting the corpus callosum (which can sometimes be done to treat sever epilepsy) and after a brain injury to one side (maybe injuring Broca’s area) there will be different side effects that result in people acting differently but to my knowledge no one is born right or left-brained and it has not been shown in any of the literature that I am aware of.
If your question refers to the fact that there are some people who rely more on the right brain and others on their left brain and that this leads to differences in behaviour, personality, etc. I am not aware of any evidence to support this.
There are however, as Dana says, differences between the hemispheres that can be apparent when some part of the brain is damaged. For example, in right-handed individuals, processing for certain aspects of language tends to be mainly done in the left hemisphere, while in left-handed individuals, it tends to be the reverse. But this doesn’t lead to any apparent differences in behaviour.
Personally, I am no specialist in the story of left-right distinctions, but think the neuromyths view (http://tinyurl.com/ow78ujm) – whilst stated rather strongly – is probably an accurate view of the neuroscientific consensus.
Of course, consensus is less important that evidence and argument. A leading supporter of the view that left-right are important is Ian McGilchrist, who wrote The Master and his Emissary. You can get get to the centre of his view, and the response of many other interesting people, here: http://tinyurl.com/3d7pwxu
Somewhat tangentially, I have never been a big fan of male-female differences either. Whilst I accept that there could be some areas in which there are important differences, I think it would be more generally interesting to understand humans as a group first!