Given a smooth variety $X$ and two smooth subvarieties $Y_{1},Y_{2} \subset X$ with smooth intersection $Z=Y_{1} \cap Y_{2}$, I'd like to compute $\mathcal{Ext}^{i}_{X}(\mathcal{O}_{Y_{1}},\mathcal{O}_{Y_{2}})$. Something that I am reading seems to suggest that the answer should be $\Omega^{i}_Z$, or at least something in terms of the normal bundle of $Z$ in $X$. But I don't see this myself.

If $c$ is the codimension of $Z$ in $Y_{2}$, then $$\mathcal{E}xt^{i}_{X}(\mathcal{O}_{Y_{1}},\mathcal{O}_{Y_{2}}) = \wedge^{c} N_{Z/Y_{2}}\otimes \wedge^{i c} \left( N_{Z/X}/N_{Z/Y_{2}}\right).$$ You can see this by Homing one Koszul complex into another. You can find the calculation e.g. in Appendix A.3 of this paper of Katz and Sharpe. 

