This suggests a variant on Polya's orchard problem.
That problem asks^{1}
for which radius $\epsilon$ of trees at each lattice point within a distance $R$ of the origin block all lines of sight to the exterior of the orchard.
It was established in the 1980's that rays to infinity are completely blocked iff
$\epsilon \ge 1/\sqrt{R^2 + 1}$, when $R$ is an integer.^{2}
E.g., for $R=20$, $\epsilon = 1/\sqrt{401} \approx 0.05$ suffices.

The variant is: Only lattice points both of whose coordinates are prime numbers are fattened to $\epsilon$-trees, and the source of visibility is $(2,2)$ rather than $(0,0)$. For example, the below shows that $\epsilon=0.3$ is insufficient to block visibility for $R=20$:

The basic question is:

. For a given $R$, which $\epsilon$ suffices to block visibility from $(2,2)$ to beyond $R$ ?Q1

As this does not seem easy to answer precisely, let me pose a very specific question:

. As $R \to \infty$, does $\epsilon \to 0$?Q2

It could be that the thinning of the prime trees is rapid enough to require some lower bound on $\epsilon$ to block visibility, even for large $R$.

^{1}Here I am quoting from a previous MO question.

^{2}Thomas T. Allen, "Polya's orchard problem." (Jstor link.)

*The American Mathematical Monthly*93(2): 98-104 (1986).)