Apparently all conjoined twins are identical twins. That means that the tissues throughout have identical genetics.
Although that means that the liver of one would be indistinguishable from the liver of the other, it may well be that the complex physiology of the twins results in many combinations of form, function and utility.
For example, the right leg is controlled by the right twin and vice versa. I presume that the sensation of touch is similarly divided.
Although the chest cavity itself appears to be shared, obviously the left twin's head is positioned less advantageously. This could suggest an asymmetrical sharing arrangement for the chest cavity below, possibly suggesting that some of the chest structures were originally part of the right twin.
The same thing may well apply to the reproductive organs. It may be that one twin or the other originally produced those organs and that the nerves serving them may not be symmetrical. If this is so then it may be that the twins will not be "sharing" one sex life.
Another example might be the lungs. It is said that there are three lungs. I don't think that automatically suggests that the twins are "sharing" one of the lungs. When the twins originally formed, it may be that the third lung belonged to one twin or the other and the present structure may not be symmetrical.
The Wiki article on them suggests that there are four lungs. " ... 4 lungs with the medial lungs moderately fused, not involving Brittany's upper right lobe". Note how the article describes the asymmetrical structure and origin of the "fused" lungs.
The comments about the diaphragm and automatic breathing is interesting. If there were two signals to the diaphragm, with the two twins opposing each other, then that would make for very ineffective breathing. Apparently that is not a problem.