Let's say there are two points on the real line, $x_1$ and $x_2$. We know the distance between them: $d=x_2-x_1$.
We obtain estimates $\tilde{x}_1$ and $\tilde{x}_2$, sampled independently from normal distributions with known variance $\sigma$, and with means $x_1$ and $x_2$ respectively.
What is the best estimate for $x_1$ and $x_2$?
It looks as if what was intended was a single observation from each distribution, and one knows which came from which distribution. That makes the problem really simple: \begin{align} \widetilde{\,x\,}_1 & \sim N(x_1,\sigma^2) \\ \widetilde{\,x\,}_2 & \sim N(x_2,\sigma^2) = N(x_1+d,\sigma^2) \\[12pt] \text{Therefore} \widetilde{\,x\,}_2 -d & \sim N(x_1,\sigma^2) \quad \text{(Recall that $d$ is known.)} \end{align} So $$ \widetilde{\,x\,}_1,(\widetilde{\,x\,}_2 -d) \sim \operatorname{i.i.d.} N(x_1,\sigma^2). $$ Therefore the mean of these two observations from that one normal distribution is a reasonable estimator of the population mean $x_1.$
Bayes's rule and Bayesianism are nowhere involved in this, although one could have a prior on $x_1$ and seek a posterior distribution.