If $X=\operatorname{Spec} A$, where $A$ is a noetherien, complete local ring, with a finite residual field $\mathbb{F}\_p$. We can associate to $A$ a rigid analytic space with two different ways, we can use Berthelot functor or the method of Raynaud. These methods rise to same rigid space when the scheme is proper over $W(\mathbb{F}\_p)$. 


If $\mathfrak{p} \in \operatorname{Spec}A(\mathbb{C}\_p)$, can we compare the completed local ring of $\operatorname{Spec}A$ and the completed local ring of the rigid analytification $(\operatorname{Spec}A)^{rig}$ of the affine scheme $\operatorname{Spec}A$? 


Can we compare the global sections $H^{0}((\operatorname{Spec}A)^{rig},\mathcal{O}\_{(\operatorname{Spec}A)^{rig}})$ and the ring $A$?



Berthelot functor, rigid analytic space
Let me say something related to your first question (the local one) in a slightly different setting. Let $\mathscr{A}$ be a $\mathbb{Q}\_p$-affinoid algebra and consider its reduction $\tilde{\mathscr{A}} = \mathscr{A}^\circ/\mathscr{A}^{\circ\circ}$. (The ring $\mathscr{A}^\circ$ is close to your $A$.) We have a reduction map $r \colon X \to \tilde{X}$, where $X$ denotes the rigid spectrum of $\mathscr{A}$ and $\tilde{X}$ the spectrum of $\tilde{\mathscr{A}}$ (a scheme).


Let $x$ be a rigid point of $X$, i.e. a point in a finite extension of $\mathbb{Q}\_p$. In the case where $\mathscr{A}$ is distinguished (in your case, it is equivalent to requiring that $\mathscr{A}$ is reduced and $\|\mathscr{A}\|\_{sup} = |k|$) and $X$ is equidimensional, S. Bosch proved that the completed local ring at the point $r(x)$ is isomorphic to the ring of functions that are bounded by 1 on the tube of $r(x)$, i.e. $r^{-1}(r(x))$ (which is an open set containing $x$).


In case you are interested, S. Bosch's result holds over an arbitrary non-archimedean complete valued with non-trivial valuation. In the discretely valued case, F. Martin reproved it in arxiv.org/abs/1510.01178 (using de Jong's results as in nfdc23's answer) and removed the equidimensionality assumption.


