Let $\iota_A:A\hookrightarrow X$ and $\iota_B:B\hookrightarrow X$ be subschemes of a smooth ambient variety $X$. Then the derived tensor product $$\mathcal O_A\stackrel{L}{\otimes}\mathcal O_B\in D^b(X)$$ is a pushforward from $A$ (of $L\iota_A^*\mathcal O_B$), and a pushforward from $B$ (of $L\iota_B^*\mathcal O_A$).

Is it also a pushforward from $A\cap B$ ?

I assume not, which is presumably the need for all the homotopy complications of derived algebraic geometry ? (By this I mean that if it is not such a pushforward, then one has to carry round some of the information of the embedding of $X$, which in derived algebraic geometry is a non-canonical local choice; finding a category in which these choices "glue" is then the nasty bit I presume, though I know $\le0$ about it.)