If $X$ is any scheme over $k$ then we know that the image of the diagonal $\Delta(X)$ is locally closed in $X \times_k X$, so that there is an open set $W$ of $X \times_k X$ with $\Delta(X)$ closed in $W$. Then we get an ideal sheaf $\mathcal{I}$ of $\mathcal{O}_W$ which defines $\Delta(X)$ as a closed subvariety of $W$. We can use this to define the formal completion of the diagonal, defined as the ringed space $$ \left(\Delta(X), \varprojlim_{n \in \mathbb{N}}(\mathcal{O}_W / \mathcal{I}^{n+1}) \right) $$ which is what I'm interested in.
Whats bugging me is that if we chose a different open set $W'$ we would get a different ideal sheaf $\mathcal{J}$ of $\mathcal{O}_{W'}$ and ultimately taking the completion of the diagonal in $W$ might be different to the completion in $W'$. Is there a way to get around this, without assuming $X$ is separated?