How do I figure out stellar distortion resulting from binary proximity?

For really, really close orbiting (non-contact) binary stars, how do I figure out how much binary stars affect or distort their companion's physical shape as a result of their masses and proximity?

Tricky.

Easiest case is to represent the stars as point masses surrounded by massless fluid, for which you can use the properties of the Roche lobe. To model it properly you need to take into account the internal structure of both stars. Probably there are some fairly reasonable approximations you can use though.