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So you have 2 overlapping paths with a heavy black stroke, and want to remove the strokes and anything behind the strokes.

Let's call your 2 paths P1 and P2.

Select them both and duplicate (ctrl-D), to produce duplicates P1d and P2d.
Keep these duplicates selected and call Path -> Stroke to path. This will convert the strokes of P1d and P2d into black hollow paths.
Select P2d and duplicate it; we'll need it twice. Now you have P1, P2, P1d, P2d and P2dd.
We don't want strokes on the final figures, so select P1 and P2 and remove the stroke (stroke width 0, or no stroke paint).
Select P1 and P1d, and call Path -> Difference (ctrl--). This will remove the area used by the black frame P1d from P1.
Select P2 and P2d, call Path -> Difference. This will remove the area of the black frame from P2.
Select P1 and P2dd, call Path -> Difference. This will remove P2's black frame from P1.

Now you are done. You may want to group or combine the 2 resulting paths into a group or a single path respectively.

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