It doesn't have specific modules for 'Left Hand', 'Right Hand', etc. Rather, it takes in information and makes sense out of it. It does this even when the setup is haphazard (as the connection between the twins' brains must be). On the other hand, we know the brain does have specific modules (such as the visual cortex among many others), which makes an interesting dichotomy.

This depends on how you interpret the term "module". One could say that once the brain starts to receive a specific type of information, it begins to form a module for that type of information.

Note that the notions of "modularity" and "adapts to environmental inputs" are not mutually exclusive in any way. As an analogy, consider embryo development. An embryo starts out as just a single cell, which then divides into two, the two of which divide into four, and so on. Gradually the cells begin to specialize in various directions, their development guided by the chemical cues released by the surrounding cells. The cells in the developing fetus / embryo respond very strongly to environmental inputs in the form of chemical cues from the other cells. In fact, without those cues, the cells would never find their right form. If those environmental cues direct the cells' development in the right direction, it will lead to the development of a highly modularized system of organs with a heart, liver, lungs, and so on. If the environmental cues are disrupted, the embryo will not develop correctly.

Now consider the brain. Like with other organs, we start off with a pretty unspecialized and general system. Over time, various parts of it grow increasingly specialized as a result of external outputs. Here external outputs are to be understood both as sense data coming from outside the brain, and the data that the surrounding parts of the brain are feeding the developing part. If the part receives the inputs that it has evolved to receive, then there's no reason why it couldn't develop increasingly specialized modules as a response to that input. On the other hand, if it doesn't receive the right inputs during the right parts of its development, the necessary cues needed to push it in a specific direction will be missing. As a result, it might never develop that functionality.

Obviously, the kinds of environmental inputs that a brain's development should be expected to depend on are the ones that have been the most consistently recurring ones during our evolution.

All of that being said, it should be obvious that "the brain takes in information and makes sense out of it" does not imply "the brain doesn't have specific modules for 'Left Hand', 'Right Hand', etc". In individuals who have developed in an ordinary fashion, without receiving extra neural inputs from a conjoined twin, the brain might have developed specific modules for moving various parts of the body. In individuals who have unexpectedly had a neural link to another brain, different kinds of modules may have developed, as the neural development was driven by different inputs.