Illustrating the burden of proof - pay me!

[Angrboda]

How do you meet in the middle when deciding who has the burden of proof for the claim that both share the burden of proof?

It's not the middle, but I think it's helpful to consider who is making the positive claim. If one side would be put in the position of trying to prove a universal negative, they don't really have the burden of proof, even if they brought it up first. It's undertood that their negative claim is a reaction to some other positive claim. If someone asserts that there are no such things as magical leprechauns, it's understood that the statement is a response to the idea that there ARE such things as magical leprechauns, even if no one in that particular conversation asserted that there are. The person you're talking to may not have claimed there are, but if they're going to disagree with the proposition, it's on them to demonstrate sufficient reason to believe leprechauns are real. As a matter of courtesy, you probably shouldn't assert there are no leprechauns apropos of nothing without being prepared to explain your reasoning if someone disagrees, but in the absence of contrary evidence, that's the default position no matter who brings it up first.

If it's not a universal negative, the person who asserts it (there's no visible elephant in my garage) has the burden of proof.

My remark was a bit tongue-in-cheek. While it's true that universal negatives may simply be an inartful response to a positive claim, it needs to be borne in mind that there are universal negatives that do bear a burden of proof. For example, if one were to argue that the bacterial flagellum could not evolve naturally, that would be a universal negative which bears a burden of proof. Otherwise, universal negatives which should be challenged are inappropriately given a pass. (A couple other examples to keep things clear, the claims that, "the second law of thermodynamic cannot be violated" and "a cause must precede its effect" implicitly contain universal negatives which would require support if they are critical premises in an argument.)