Not sure what to say in general, but in practice one often uses a semismall resolution. For example if your sheaf $IC(L)$ is given as $\pi_* C_Z$ (pushforward of constant sheaf, or more generally orientation or dualizing sheaf if - I'm assuming $Z$ is not smooth and oriented) for a proper $Z\to X$ you can calculate derived self-maps $$Hom(\pi_*C_Z,\pi_*C_Z)= Hom(C_Z, \pi^!pi_*C_Z).$$$$Hom(\pi_*C_Z,\pi_*C_Z)= Hom(C_Z, \pi^!\pi_*C_Z).$$ Now write $\pi_1,\pi_2$ for the two projections from $Z\times_X Z\to Z$. Then we calculate further
$$Hom(C_Z, \pi^!pi_*C_Z)=R\Gamma_Z(\pi_{1,*}\pi_2^!C_Z)=R\Gamma_{Z\times_X Z}(\omega_{Z\times_X Z}),$$$$Hom(C_Z, \pi^!\pi_*C_Z)=R\Gamma_Z(\pi_{1,*}\pi_2^!C_Z)=R\Gamma_{Z\times_X Z}(\omega_{Z\times_X Z}),$$
i.e., Borel-Moore homology of $Z\times_X Z$ (theup to a shift I've ignored). This works in the equivariant setting (i.e. for sheaves on stacks) equally well, giving equivariant Borel-Moore homology. The famous example of this beingis the case where $X$ is the nilpotent cone [or its quotient by the group], $Z$ the [equivariant] Springer resolution, and $Z\times_X Z$ the [equivariant] Steinberg variety, cf. the book of Chriss-Ginzburg. In this case the equivariant BM homology (=Ext algebra of the Springer sheaf) is the degenerate affine Hecke algebra. This calculation works as is in the equivariantdg setting (i.e.--- for sheavesthe statement on stacks) equally wellthe level of derived categories you need formality, giving equivariant Borel-Moore homologywhich in the Springer case is a result of Laura Rider here.