Filling in overlapping areas in Draw

I want to make classic logic Venn diagrams. This requires me to fill in overlapping areas. As far as I can tell, I can only fill in entire shapes. I want merely to fill in certain areas of overlap with color. Is this possible?

You are correct about only being able to fill entire shapes.

There is an extension available, which will give you the result below.

image description

Or you can draw/fill the shapes and adjust the transparency.