The EEG signals between these regions will not be independent. The electrical activity that is being measured at the scalp, is the result of electrical flow inside the brain. This flow does not go in just one direction. It 'smears out' a little bit to surrounding areas, since the entire scalp/body is conductive.
Another reason that the activity measured at different sites is not completely independent is because brain areas interact with each other. The old Brodman Area topography is kind of outdated, and made place for networks of activity. To say that each channel is independent is thus inherently false.
ICA does assume independent sources, but with this they do not mean the channels. For example, the ICA finds a pattern, e.g. jumps of blinks, in, say, 15 channels. ICA then assumes that they are caused by only one thing, and this pattern will thus be extracted to only one component, not two. The different components, which are the result of an ICA, are thus in fact the independent sources.