This paper gives a direct proof of localization of dual norms of bounded linear functionals on the Sobolev space $W^{1,p}_0(\Omega)$, $1 \leq p \leq \infty$. The basic condition is that the functional in question vanishes over locally supported test functions from $W^{1,p}_0(\Omega)$ which form a partition of unity in $\Omega$, apart from close to the boundary $\partial \Omega$. We also study how to weaken this condition. The results allow in particular to establish local efficiency and robustness with respect to the exponent $p$ of a posteriori estimates for nonlinear partial differential equations in divergence form, including the case of inexact solvers. Numerical illustrations support the theory.