I don't see how you'd be guaranteed a simple sample of $D_1$ by subtracting points in $D_2$ from different points in $D$. For example, consider these distributions:
If you happen to randomly sample the first five values in a row (it could happen!) from $D$ and subtract five randomly sampled values from $D_2$, you're pretty unlikely to end up with a sample of $D_1$; you'll probably get a bunch of negative values, none of which belong to $D_1$ of course.
I.e., I think you'd only get to sample randomly from $D$ or $D_2$, not both. Same goes for stratified sampling: it'll only work if you sample systematically within strata, not randomly – at least, not randomly with both samples. E.g., you could sample randomly from $D$, with or without stratification, but if you happen to sample the 8th, 48th, and 88th values from $D$, the only way I can see to guarantee that you'll get values from $D_1$ by subtracting values sampled from $D_2$ is to systematically sample the same 8th, 48th, and 88th values from $D_2$.