Each side of the brain handles specific functions. It may be that these reactions we see in split brain patients are the right brain coping with it’s sudden inability to express itself as part of the cohesive whole of the individual. Consider when you do something like typing, the right and left brains have to work in tandem (at least if you’re typing with two hands.) This means that one side of the brain is determining what to write, communicating that intent to the other side of the brain, and then working -with- that other side to perform the physical act of typing. This indicates that on a basic level, the two sides of your brain work together. In cases where the brain is split, the non-verbal half of your brain does not suddenly stop thinking. What’s happened is that it’s ability to communicate that thought has been removed. The reaction we see in split brain people may be the right brain’s coping mechanism attempting to re-establish the communication capacity it lost after being disconnected from the left brain. The disagreements that arise are no different from the internal disagreement you have when picking out clothes in the morning, except instead of handling them internally, like a non-split brain would, the “discussion” has to take place physically.
There’s nothing in this to indicate that you are two people, only that each half of your brain makes decisions based on what it perceives to be important. That’s nothing new to science.