In a game from the following paper, it is stated that
Player $i$ observes a private signal $x_i = \theta + \epsilon_i$. Each $\epsilon_i$ is independently normally distributed with mean $0$ and standard deviation $\sigma$. We assume that $\theta$ is randomly drawn from the real line, with each realization equally likely. This implies that a player observing signal $x$ considers $\theta$ to be distributed normally with mean $x$ and standard deviation $\sigma$. This in turn implies that he thinks his opponent’s signal $x$ is normally distributed with mean $x$ and standard deviation $\sqrt{2}\sigma$. The assumption that $\theta$ is uniformly distributed on the real line is nonstandard, but presents no technical difficulties
Can someone please help me understand why $x_1\mid x_2\sim N(x_2,2\sigma^2)$?
$$ f_{X_1,X_2,\theta}(x_1,x_2,t)=f_{\varepsilon_1}(x_1-t)\cdot f_{\varepsilon_2}(x_2-t)\cdot f_\theta(t)$$
"integrate" over $t$ using $f_\theta(t)=1$ to get $$f_{X_1,X_2}(x_1,x_2)$$ proportional to $$ \exp\left(-(x_1-x_2)^2/(4\sigma^2)\right)$$
do a similar reasoning to get $f_{X_2}$ and then use Bayes' rule.