From an evolutionary standpoint, the main reason why you'd have separate brains or a decentralized brain as in the case of the octopus is to dedicate "brain" to certain actions without detracting from other processes. In other words, the central brain of an octopus would likely not have to think about how to grapple its prey, only that it wants to grapple its prey, and the nerve cells in its tenticles make the individual detailed decisions regarding how it could grapple its prey.
You could think of it as having a graphics processing core like most modern graphics cards have nowadays. While the CPU could perform the same calculations as the GPU, due to the quantity of calculations that must be performed, it is more efficient to specialize these calculations. And like this metaphor, the GPU doesn't act on its own accord, but rather it is the CPU that tells it to act. In the same way the nerve cells in the tenticles of an octopus specialize in dealing with how the tenticle should move.
It is difficult to imagine, but in a certain sense we do it too. The part of our brain that deals largely with automated actions like breathing and blinking our eyes is closer to our spinal cord and as such, those signals rarely arrive to our waking brain. Does this mean we can't control when we breathe or when we blink our eyes? No, however, to a certain extent, you could think of these functionalities as being specialized in order to not detract from thinking about other things. You may not realize, but the reason why you come up with ideas much more readily on the toilet is because your brain is not preoccupied with standing and, more importantly, balance. Thus you have more brain power to think about other things.
I'm thinking of an octopus like creature that used 8, or so, separate
"brains" in concert. Each "brain" would be capable of maintaining the
creatures basic bodily functions (cardiovascular, respiratory,
digestive, and so on...), as well as being able to function
independently or cooperatively with the other "brains".
Would such a creature design be feasible/believable for an animal
It wouldn't make much sense to have a separate brain just for cardiovascular, respiratory, digestive, etc. Monotonous tasks like breathing and beating of your heart don't require much "brain" to do. Based on what you find in nature, it is far more likely that you would have separate brain to govern movement and actions, which can get quite complex. Also, if one of these separate brains were in a less-protected part of the body and got injured, then if it handles movement of the limb, you could still survive. If it handled cardiovascular activity, you're dead. So perhaps there is also a good reason in terms of survivability that this tends to be what happens in nature.
Taking this a step further, if we were to take this creature and
anthropomorphize it a bit, would it appear to have some sort of
dissociative identity disorder (split personality)?
This, you don't see anywhere in nature, so we have no frame of reference. However it is also true that we can't just put them on a psychologist's couch and ask them what they're thinking, so we also can't know for sure. However speculating, we could look at ourselves. While it's true that we don't have truly separate brains, we have two hemispheres that work in synch to solve problems. The left brain tends to dominate in most people, and concerns itself with logic and detail while the right brain tends to deal with aspect and overall appearance (though recent studies suggest that this difference isn't as strong as initially thought).
The reason behind the fact that most people in their adult years still draw stick figures is due to the fact that the left brain, not knowing how to draw a person but thinking it knows anyway, forces control over the right brain which otherwise would have carefully and patiently studied and remembered how a person looks in order to draw effectively. What feels like you drawing a stick figure is actually your left brain saying, "Hey, back off, right brain, I know how to do this.. just make a circle and a stick underneath with a line for the arms and two lines for the legs.. DONE!" In fact the first step to learning how to draw well is dropping preconceptions about how to draw things and just trying to draw what you see. You may still have difficulty drawing doing so, but only because your right brain is practicing for the first time.
If our hemispheres were separate, due to the distance to send signals, it may not seem so much like you but more like your left and your right. This is just speculating, but our identity too might get split somewhat. However remember that evolution won't support a creature that can't "work," so despite this conflict in identity, they would have to get along well (no two-headed ogre stuff or nothing would get done).
Would each decision would have to be run through committee or would it
make more sense to have each brain take control from time to time?
Returning back to the discussion of drawing, in that instance, the left brain just takes over. There is no consensus, and in a certain sense, there shouldn't be any, or you wouldnt' get anything done. Certain pathways tend to allow one hemisphere to have an edge over the other.
However this isn't to say that your hemispheres don't battle certain decisions out. The Rubber Hand Experiment is a classic example of such conflict. Your left brain knows the hand is fake, but your right brain sees and feels something different. When danger strikes, you're forced to come to terms with which hemisphere is right. Even though you know otherwise, you still get the same response as you would if someone were to try to hammer your real hand, because one of your hemispheres believed it were real. Usually when confronted with such a conflict, the signal that claims danger is the one that dominates (and for good reason). So to answer your question, likely each brain would formulate its own opinion on the matter and the brain with the strongest opinion would probably win over the others. However this is all very much speculative.
Check here for recommended reading on the subject. I hope that helps.