> We also have a nervous system distributed throughout our bodies, so we do experience the full body, but the neural signals are sent to one location, not nine.

Mostly yes. But we do have some processing that bypasses the brain: reflexes for example. The classic example is the kick reflex that doctors test with a small mallet on your knee. Or when you blink as something flies towards your eyes. It happens and your brain gets informed afterwards.

But there are degrees of non-conscious processing, from the heart beating by itself and your guts doing their things, to the breath which will be on autopilot until you take control over it, to entirely voluntary things like moving about your limbs (for healthy individuals).

Maybe the octopuses experience their disconnected brains somewhere on that spectra, where the limbs are semi-autonomous unless it focuses on them? (All we can do is speculate, there is no known way to learn the qualia of being an octupus.)

Other Minds: The Octopus, the Sea, and the Deep Origins of Consciousness [excellent]

https://www.amazon.com/Other-Minds-Octopus-Origins-Conscious...

If you haven't read it, you might enjoy Adrian Tchaikovsky's Children of Time series. It's fiction which explores different intelligences, with the second novel[0] dealing specifically with octopuses.

[0] - https://en.wikipedia.org/wiki/Children_of_Ruin

While the specifics of octopus experience are hard for us to fathom, we do say our stomachs have a mind of their own, and science has backed that up mostly. The enteric nervous system is complex and large and intimately in touch with gut microbes (of which there are trillions), and is connected to the brain. Both speak to each other, but the gut does a lot of independent stuff, and we’re learning a lot of the chemicals the gut microbiome releases shapes our moods and behavior.

For an octopus, it must be something similar but with all 8 of their appendages. Some specialize for hunting vs moving, so that’s at least easy to understand, but given they can act independently, I wonder if the octopus feels like the conflict between gut and brain with respect to food.

Mind != Consciousness

How sure are you that you have a unified embodied experience? Have you ever caught an unexpected ball? Run (I mean, really run)? There are a lot of situations where you conscious self isn't in charge so much as it is a really good storyteller convincing 'you' that you've done something yourself that you had no part in. Cut the interface between brain hemispheres and 'you' will still have the absolute self assurance that 'you' are the one in charge.