This is an elaboration of Lucia's comment. Let us consider the Dirichlet coefficients of $L(s,\pi\times\pi')$, $L(s,\pi\times\tilde\pi)$, $L(s,\pi'\times\tilde\pi')$ at a prime power $p^k$. Following the proof of Proposition 6 in Molteni's paper (Duke Math. J. 111 (2002), 133-158), we see that there exists a polynomial $P_k\in\mathbb{N}[x_1,\dots,x_k]$ such that
$$ k!\ a_{\pi\times\pi'}(p^k) = P_k(\sigma_1\tau_1,\dots,\sigma_k\tau_k),$$
$$ k!\ a_{\pi\times\tilde\pi}(p^k) = P_k(|\sigma_1|^2,\dots,|\sigma_k|^2),$$
$$ k!\ a_{\pi'\times\tilde\pi'}(p^k) = P_k(|\tau_1|^2,\dots,|\tau_k|^2),$$
where $\sigma_h$ (resp. $\tau_h$) denotes the $h$-th power sum of the Langlands parameters of $\pi$ (resp. $\pi'$) at $p$. Using Cauchy-Schwarz, it follows that
$$ |a_{\pi\times\pi'}(p^k)|\leq \sqrt{a_{\pi\times\tilde\pi}(p^k)a_{\pi'\times\tilde\pi'}(p^k)}. $$
Note that the coefficients on the right are nonnegative. Hence for all positive integers $n$, we have
$$ |a_{\pi\times\pi'}(n)|\leq \sqrt{a_{\pi\times\tilde\pi}(n)a_{\pi'\times\tilde\pi'}(n)}\leq (a_{\pi\times\tilde\pi}(n)+a_{\pi'\times\tilde\pi'}(n))/2.$$
It follows that the absolute Dirichlet series for $L(s,\pi\times\pi')$ is majorized by the average of the Dirichlet series for $L(\sigma,\pi\times\tilde\pi)$ and $L(\sigma,\pi'\times\tilde\pi')$. The latter converges for $\sigma>1$ by Landau's lemma, hence $L(s,\pi\times\pi')$ converges absolutely for $\Re(s)=\sigma>1$.

**Added 1.** The above argument requires that $p$ is an unramified prime for $\pi$ and $\pi'$, hence also that $n$ is only divisible by such primes. This does not alter the final conclusion though, because omitting or including finitely many Euler factors does not alter absolute convergence of the corresponding Dirichlet series in the half-plane $\Re(s)>1$.

**Added 2.** For a related and more complete result concerning the logarithmic derivative of Rankin-Selberg $L$-functions, see Brumley's Appendix to the exciting new preprint of Soundararajan and Thorner.