There are well known techniques for proving lower bounds on the communication complexity of boolean functions, like fooling sets, the rank of the communication matrix, and discepancy.
1) How do we use these techniques for lower bounding partial boolean functions? More specifically, how do you count the rectangles in the communication matrix? Do you also count the "undefined part" of the function? Or do you leave it out?
Another thing, what are the known relations between the communication complexity of total and partial functions? For example, is there a function $f$ with a promise version of it, call it $f'$, such that $C(f)\neq C(f')$? (here $C$ could be any communication complexity measure like deterministic, probabilistic with all its flavors, quantum). A concrete example is the equality function EQ, and its promise version defined in this paper denoted as EQ'. It is known that $C(EQ)=n$ and $C(EQ')=\Omega(n)$ where $C$ is the bounded error communication complexity (see the paper). There is no matching upper bound here, but they are asymptotically the same.
2) Is there a function defined in the same spirit of EQ and EQ', but with a different complexity?