  For a given shape $S$ in the plane, one can ask what is the lowest possible
density of a point set $P$ that pierces ("intersects", "hits") all translates
of $S$. This is equivalent to determining the covering density of $S$ and as
such is well studied. Here we study the analogous question for families of
shapes where the connection to covering no longer exists. That is, we require
that a single point set $P$ simultaneously pierces each translate of each shape
from some family $\mathcal F$. We denote the lowest possible density of such an
$\mathcal F$-piercing point set by $\pi_T(\mathcal F)$. Specifically, we focus
on families $\mathcal F$ consisting of axis-parallel rectangles. When
$|\mathcal F|=2$ we exactly solve the case when one rectangle is more squarish
than $2\times 1$, and give bounds (within $10\,\%$ of each other) for the
remaining case when one rectangle is wide and the other one is tall. When
$|\mathcal F|\ge 2$ we present a linear-time constant-factor approximation
algorithm for computing $\pi_T(\mathcal F)$ (with ratio $1.895$).
