EDIT If you have never used a Bayesian method before, you should seek detailed support from someone you work with. I state this because your comment has the feel of someone who hasn't done it before.
A Bayesian prior summarizes all of your knowledge about the location of the parameters in a model, not just $\beta$. In this case, if $\tilde{\mathbf{x}}$ is being treated as a parameter, then it also needs a prior.
A prior is proper if $$\int_{\theta\in\Theta}\pi(\theta)\mathrm{d}\theta=1.$$ It is improper if that statement is not true. Frequentist solutions usually map to a Bayesian solution with an improper prior. The challenge is that for a normal likelihood, when you have three or more independent variables, then an improper prior will cause the posterior to not integrate to one. You will get paradoxes in your solution.
The prior comes from information outside the sample. You would assign a prior for each parameter on the full data set. The difference between Bayesian and Frequentist methods is that in Frequentist methods the sample is random. In Bayesian methods, the parameters are the random variables. The sample is treated as a constant. For the full set, you could assign a normal-gamma or normal-inverse-gamma distribution. For the partial data set, you would use the posterior of the full set as the prior of the new set, but add one dimension, your believed distribution for $\tilde{\mathbf{x}}.$
You are correct, you would capture your prior from the $\mathbf{x}_i$, but it wouldn't be a simulation. As you observed more and more data from outside the full set, your shape will change. Each observation will cause Bayesian updating which will change the distribution of $\mathbf{x}_i$.