Let $C$ be a smooth proper curve over an algebraically closed field $k$ with $p,q \in C$ two distinct points. Let $X = C / p \sim q$ be the nodal curve obtained by gluing $p$ and $q$. Denote the node by $n \in X$ and the normalization map by $\pi : C \to X$.
We can relate the functions on $X$ with those on $C$ by the exact sequence: $$ 0 \to \mathcal{O}_X \to \pi_*\mathcal{O}_C \overset{\rho}{\to} k \to 0 $$ where $\rho(f) = f(p)-f(q)$. In other words, the functions on $X$ are functions on $C$ which assume the same value on $p$ and $q$.
Let $A = \mathcal{O}_{X,n}$ and $B = (\pi_*\mathcal{O}_C)_n = \mathcal{O}_{C,p\cup q}$. It is a standard fact that $\hat A \simeq k[[x,y]]/(xy)$.
Could we choose $x$ and $y$ as local analytic functions coming from $C$ (possibly defined near $p$ and $q$)?
Moreover, by classification of torsion free sheaves we must have $B\otimes_A \hat A \simeq \langle f_1,f_2 | yf_1=xf_2 = 0 \rangle$. So we ask again:
Could we choose $f_1$ and $f_2$ to be local analytic functions defined on $C$?
By a local analytic function near $p \in C$, I mean an element of $\hat{\mathcal{O}}_{C,p}$. Therefore, I'm looking for explicit isomorphisms where $x,y$ and $f_1,f_2$ are somewhat natural.
I am not sure I have understood your question correctly, but here goes.
$B\otimes_A\hat{A}=\widehat{\mathcal{O}_{C,p}}\times\widehat{\mathcal{O}_{C,q}}$ and the latter is just $k[[x]]\times k[[y]]$, where $x,y$ are the local analytic parameters at $p,q$. You have the natural map to $k$ from this ring, given by $(f(x),g(y))\mapsto f(0)-g(0)$. The kernel is $\hat{A}$. I hope this clarifies the situation.