Completed Tensorproduct

I am trying to understand the completed tensorproduct. This can be defined as follows: Given a topological ring $R$ and two linearly topologized rings $A$ and $B$ with fundamental systems of open subsets $A_\mu$ and $B_\nu$, we define the completed tensorproduct $A\otimes^\wedge _R B$ to be the completion of $A\otimes _RB$ with respect to the system $\text{Im}(A_\mu \otimes_R B + A\otimes_R B_\nu \rightarrow A\otimes _R B)$ (see stacksproject Tag 0AMU).

How do such tensor products look like in practice? For example, if we consider something like $k[[t]][X]$ over $k[[t]]$ how does the tensor product with itself look like? How does it differ from $k[[t]][X,Y]$?

Also, do I have a nice universal property as in the case of the "usual tensorproduct"?