Is it true that $$C^*_r(G_1\times G_2)=C^*_r(G_1)\otimes_{\min}C^*_r(G_2)$$ for locally compact groups $G_1$ and $G_2$? I have managed to prove that it holds for discrete groups (see below), but as far as I can see, in order to expand the result to locally compact groups, we need the extra assumption of second-countability.
Let $\lambda_1$ and $\lambda_2$ denote the left-regular representations of $\Gamma_1$ and $\Gamma_2$ respectively, and let $\lambda$ be the left-regular representation of $\Gamma_1\times \Gamma_2$. The minimal tensor product $C^*_r(\Gamma_1)\otimes_{\min}C^*_r(\Gamma_2)$ is the norm closure of the linear span of the subset $$\left\{S\otimes T\,|\,S\in C^*_r(\Gamma_1),\ T\in C^*_r(\Gamma_2)\right\}\subseteq B(\ell^2(\Gamma_1)\otimes\ell^2(\Gamma_2)).$$ We identify the Hilbert spaces $\ell^2(\Gamma_1\times\Gamma_2)$ and $\ell^2(\Gamma_1)\otimes\ell^2(\Gamma_2)$ by means of the unitary operator $U$ that satisfies $$U(\delta_{(s,t)})=\delta_s\otimes\delta_t,\quad s\in\Gamma_1,\ t\in\Gamma_2.$$ Under this identification, it is easy to see that $\lambda(s,t)=\lambda_1(s)\otimes\lambda_2(t)$ for all $s\in\Gamma_1$ and $t\in\Gamma_2$, from which it follows that $C^*_r(\Gamma_1\times\Gamma_2)\subseteq C^*_r(\Gamma_1)\otimes_{\min}C^*_r(\Gamma_2)$. As the complex group rings clearly satisfy $\mathbb{C}\Gamma_1\odot\mathbb{C}\Gamma_2\subseteq\mathbb{C}(\Gamma_1\times\Gamma_2)$, a density argument shows the other inclusion.