Noncoprime polynomial values

Here's an argument that might be a bit too high-brow (it uses Chebotarev's density theorem).

Reduction 1. Your question is equivalent to: for $f_1, \ldots, f_n \in \mathbb Z[x]$, does there exist a prime $p$ such that all $f_i$ have a root in $\mathbb F_p$. Indeed, taking lifts $x_i \in \mathbb Z$ of roots $\bar x_i \in \mathbb F_p$ of $\bar f_i \in \mathbb F_p[x]$ gives $$f_i(x_i) \equiv 0 \pmod p$$ for all $i$, so the $f_i(x_i)$ have $p$ as common divisor.

Reduction 2. We may assume that all $f_i$ are irreducible. Indeed, a root (mod $p$) of a factor of $f_i$ gives a root of $f_i$ itself.

Proof of the assertion. Let $K_i = \mathbb Q[x]/(f_i)$ be the field extension defined by adjoining a root of $f_i$. Let $K$ be the Galois closure of the compositum of the $K_i$ (inside $\bar {\mathbb Q}$). By the Chebotarev density theorem, there exists a prime $p$ such that $(p)$ is completely split in $K$. Thus, all the $f_i$ have a root modulo $p$. $\square$

Remark. This actually proves more: there are infinitely many primes $p$ with this property (and their density is $1/[K:\mathbb Q]$). Moreover, there are many lifts $x_i$ of a root of $\bar{f}_i$, so we actually get many examples of such $x_i$. One can ask how many there are (I will not attempt this right now).

Here is a bit more pedestrian argument. We may assume that $p_i$ are irreducible. Take $\alpha_i$ a root of $p_i$; then $\mathbb Q[\alpha_1,\dots,\alpha_n]=\mathbb Q[\xi]$ by the primitive element theorem. Taking $q$ to be the monic minimal polynomial of $\xi$, we have $\alpha_i=r_i(\xi)$, so $q(x)\mid p_i(r_i(x))$. Now we may take any prime divisor $p$ of the values of $q$ which does not appear in the denominators of the coefficients of $q$ and $r_i$; $p$ divides appropriate values of $p_i$ as well.