This is a follow-up to About the tensor product of two local rings.
Let $A, B$ and $C$ be local rings (commutative and noetherian). Suppose that we have local ring maps $C \to A$ and $C \to B$.
What assumptions guarantee that the tensor product $A \otimes_C B$ of $A$ and $B$ over $C$ is a local ring?
For example, $A$ is the completion of $C$? What if $A, B$ and $C$ are dvrs? Are there general criteria? (If I remember correctly, I have seen one criterion, when $C$ was a field. What about general $C$?)