We can very nearly solve the separable portion. The transcendental portion seems simple enough, and I'm not sure about the inseparable. This consists of most of the interesting part of the problem for me. I'm not sure what you think.

The tensor product is a field if the Galois closures of $K$ and $L$ do not contain any subfields which are isomorphic. Proof:

We can reduce to the case with $K/k$ and $L/l$ Galois, since $K \otimes L$ is contained in the tensor product of the Galois closures, and an Artin ring inside a field is a field.

Let $K/k$ and $L/k$ Galois, then $K\otimes L$ contains $k(K,L)$. We must show that the degrees of this extension is large enough. Consider the Galois group $G$, which contains subgroups $H_K$ and $H_L$ that fix $K$ and $L$. $H_K$ and $H_L$ are normal, and they are not together contained in any subgroup. Therefore $H_KH_L=G$, so $|H_K||H_L|\geq |G|$, so  $|G/H_K||G/H_L|\leq G$, so $[k(K,L):k]\geq[K:k][L:k]$, so the extension is a field.

Edit: If one of the fields is algebraic, adding transcendentals to the second one couldn't possibly make it not a field, so we only need to consider the algebraic parts of the second.