According to a Canadian television documentary the girls' brains "make them unique in the world. Their brains are connected by a thalamic bridge, connecting the thalamus of one with that of the other. The thalamus acts like a switchboard relaying sensory and motor signals and regulating consciousness.
As Klinghoffer notes, though, the title of the documentary, Twin Life: Sharing Mind and Body, is misleading because although Tatiana and Krista Hogan of British Columbia share a range of senses and can move parts of each other’s bodies, their minds, their personalities, seem to be separate. Klinghoffer writes:
Neurological studies have stunned the doctors. Tatiana can see out of both of Krista’s eyes, while Krista can only see out of one of Tatiana’s. They also share the senses of touch and taste and the connection even extends to motor control. Tatiana controls 3 arms and a leg, while Krista controls 3 legs and an arm.A materialist who believes that immaterial minds don't exist might argue that the girls don't seem to share the same "mind" because they don't entirely share the same brain. If the two girls had only a single brain between them then, the materialist could argue, they would have the same personality, etc. Moreover, the fact that they can "read each other's mind," at least partially, suggests that their "minds" are not completely separate anyway.
Amazingly, the girls say they also know one another’s thoughts without needing to speak. “We talk in our heads” is how they describe it.
But here’s the main point. Insofar as the mind subsumes the personality, these girls do not share a mind. Though together all the time, by physical necessity, their spirits are distinct, with “very different personalities.”
Despite their unique connection, the twins remain two distinct people. Tatiana is talkative, outgoing and high-strung, while Krista is quieter, more relaxed and loves to joke. But she has a temper and can be aggressive if she doesn’t get her way.
On the other hand, a proponent of mind/brain dualism - the belief that we are comprised of both a material brain and an immaterial mind - might reason that the very different nature of the girls' personalities points to something else about them that's not established, or at least not completely established, by their brains. That they share much of their brain in common with each other but possess disparate personalities suggests that something other than the brain is responsible for their personalities.
Their case is certainly compatible with the dualist view, but doesn't seem to provide conclusive evidence for it. In any event, here's the documentary of the girls made several years ago. The girls and their family are very remarkable people: