There is a huge bundle on nerves connecting the right and left hemispheres of the brain and cutting it was a technique used to control otherwise intractable seizures. Does the patient end up with two independent conscious entities inside his/her skull? ("skullmates").
A study from 1977 examined this issue when it asked questions to the left and right hemisphere independently and compared the answers. This can be done by asking written questions, and flashing them in either the left or right visual field.. from which they travel to the opposite hemisphere.
It turns out that the right hemisphere was in "a bad mood" on one occasion. Also, it liked "Dope" more, and wanted to be a "Car racer"- whereas the boring Left Hemisphere wanted to be a "Draughtsman".
Another more recent article suggests that although the 2 hemispheres cannot communicate directly, and are unaware of each other- they can share/integrate information via deeper brain structures..