This really hinges on the question of "What is the relationship of the mind/soul to the physical body?" I've heard two major ways of interpreting evidence such as the split-brain experiments. One is that the mind/soul is actually generated by the physical brain, which kind of messes with some ideas such as the immortality and divine origin of the soul. The other is that the physical brain is merely the means of communication between the immaterial soul and the physical body.
The former view, that our minds and consciousness are created by the brain, is the view most frequently accepted by scientists who study the brain, due to the obvious effect that brain injuries have on such things as personality and mental capacity. The latter view, however, is supported by some anecdotes about things such as out-of-body experiences, reincarnation, and at least one story I'm aware of of a blind woman who claims she was miraculously granted sight for a short time and was able to describe what she saw. All these incidents, if true, would suggest that our senses and consciousness are not strictly dependent on our brains and sense organs.
There is a third view, which is that the soul is indeed created by the physical brain, but then does persist after death due to physical or spiritual phenomena. There are a couple of scientists working on this who think they may have found a way that data stored in the brain could be preserved at the sub-molecular level after death, but from what I understand their work is quite speculative so far.
From my perspective personally, it actually doesn't seem odd to me to talk about conflicting opinions and desires being in the same brain. In fact, some psychologists and neuroscientists have emphasized this point; that different parts of the brain and/or mind want different things, and that these struggles generally take place within the mind of the person. If two halves of the brain are separated with no means of communication, that struggle can no longer take place internally. The two sides now both have control of half the body, and no way of communicating except through the external world.
Now it is a bit weird to imagine two independent consciousnesses arising from a brain that used to be home to a single, unified consciousness. That's the part people really can't seem to get their heads around. Is it possible to divide a single consciousness into two? Apparently. Each brain hemisphere seems to have all the equipment it needs to be independently conscious! I guess normally they are just unified because they are constantly talking to each other so much through the corpus callosum, that they have joint perceptions and reach joint decisions.
Well, there's my two cents.