  Let $p$ be prime and $X$ be a Haar-random $n \times n$ matrix over
$\mathbb{Z}_{p}$, the ring of $p$-adic integers. Let $P_{1}(t), \dots, P_{l}(t)
\in \mathbb{Z}_{p}[t]$ be monic polynomials of degree at most $2$ whose images
modulo $p$ are distinct and irreducible in $\mathbb{F}_{p}[t]$. For each $j$,
let $G_{j}$ be a finite module over $\mathbb{Z}_{p}[t]/(P_{j}(t))$. We show
that as $n$ goes to infinity, the probabilities that $\mathrm{cok}(P_{j}(X))
\simeq G_{j}$ are independent, and each probability can be described in terms
of a Cohen-Lenstra distribution. We also show that for any fixed $n$, the
probability that $\mathrm{cok}(P_{j}(X)) \simeq G_{j}$ for each $j$ is a
constant multiple of the probability that that $\mathrm{cok}(P_{j}(\bar{X}))
\simeq G_{j}/pG_{j}$ for each $j$, where $\bar{X}$ is an $n \times n$ uniformly
random matrix over $\mathbb{F}_{p}$. These results generalize work of Friedman
and Washington and prove new cases of a conjecture of Cheong and Huang.
