It seems to me like you should be able to do this pair-wise — divide first one pair of circles, then the resulting objects (which one can group) with the other circle.
EDIT: Okay, to test my idea I installed Inkscape on this machine and I see what you mean: there is not a function to "cut up" a pair of objects along the boundary of their overlap, exactly. Otherwise my idea would have worked. I tried exclusion on one pair of circles, then again with the result and a third circle, then broke them apart (Control-Shift-K, also in the Path menu) and what I got was almost what you want, but the places where two but not all circles overlapped were negative spaces rather than shapes like you mentioned was your goal (the seven separated parts). I tried to fix this with the Fill Bounded Areas tool (paintbucket, Shift-F7) but the shapes were a bit smaller than desired; the outset tool [ Control - ) ] helped but results were not perfect.
Probably there is a conceptually better way to recreate the missing three shapes by dereticulating splines (if I may coin a term with an origin borrowed from the Sims, who borrowed it from mathematics), picking just the curves (beziers, lines) you want out of each shape and recombining curves from adjacent shapes to make a new one, filling in the gap. However, I do not know how to do this.
-Arlo James Barnes