Estimating norm of sum of elements of $W^{-1,p}$ with disjoint supports

Question

Suppose an open set $\Omega \subset \mathbb{R^n}$ is partitioned into disjoint sets $\Omega_1, \Omega_2$ (make all the assumptions on geometry you want). Let $f_1, f_2$ be elements of $W^{-1,p}(\Omega_1)$ and $W^{-1,p}(\Omega_2)$ respectively, compactly supported (positive distance from the boundary) in their respective domains. I want to define $f \in W^{-1,p}(\Omega)$ as a sum of $f_1$ and $f_2$ (which is doable due to the compact support assumption). I'm wondering what's the best estimate on the $W^{-1,p}$ norm of $f$ I can get. Ideally I'd like it to be something like $||f||_{W^{-1,p}(\Omega)}^p \leq ||f_1||_{W^{-1,p}(\Omega_1)}^p + ||f_2||_{W^{-1,p}(\Omega_2)}^p$, but I'm not really sure if it holds - I just thought that that's the best I can possibly hope for, and it's the case if $f_1, f_2$ are represented by compactly supported $L^p$ functions. Would very much appreciate any suggestions for a proof/counterexample, a weaker estimate would be fine too.