We study a variant of the Localization game in which the cops have limited visibility, along with the corresponding optimization parameter, the $k$-visibility localization number $ζ_k$, where $k$ is a non-negative integer. We give bounds on $k$-visibility localization numbers related to domination, maximum degree, and isoperimetric inequalities. For all $k$, we give a family of trees with unbounded $ζ_k$ values. Extending results known for the localization number, we show that for $k\geq 2$, every tree contains a subdivision with $ζ_k = 1$. For many $n$, we give the exact value of $ζ_k$ for the $n \times n$ Cartesian grid graphs, with the remaining cases being one of two values as long as $n$ is sufficiently large. These examples also illustrate that $ζ_i \neq ζ_j$ for all distinct choices of $i$ and $j.$