The way I see it the one making the claim has the burden of proof. So if a theist makes the claim that God exists, the burden of proof is on them to support that proposition.
However the proposition that no God exists (which I think goes beyond disbelief until proportional evidence to the claim is given) also seems like a positive claim as well and needs justification/proof too even though the claim is negative.
So who has the burden of proof, one or the other, both, neither?
I think this matter can be made clearer if we rephrase the statements, "God exists" and "God does not exist" to include their hidden assumptions. Thus the statements can be rephrased accordingly:
"It is demonstrable that god exists".
"It is demonstrable that no god exists".
The first case, "It is demonstrable that god exists", seems to me an obvious rephrasing of the assertion, for the assertion that something exists logically entails that that thing can be demonstrated to exist, in so far as one is making a factual claim and is not merely taking about one's beliefs.
But some of us might not see how the second case, "It is demonstrable that no god exists" is a logical rephrasing of the assertion, "No god exists", so I will explain.
First, assume someone were to assert that "It cannot be demonstrated that any god exists". Logically, that is not the same as asserting "It can be demonstrated that no god exists". Moreover, it is the position of agnosticism, rather than atheism. So, the atheist is not making the claim that "It cannot be demonstrated any god exists" --- or more precisely, he or she is not asserting that as a truth claim --- but to the extent they are asserting it at all, they are asserting it as a belief, rather than as a truth.
Put differently, "God exists" and "No god exists" are both claims about one's beliefs, rather than claims about what is or is not true, unless they are both understood to be claims about what is or is not
demonstrably true.
Now let's return to those first two claims:
"It is demonstrable that god exists."
"It is demonstrable that no god exists."
As a rule of logic,
the burden of proof always lies with the person making a positive or affirmative claim.
So which of those propositions makes a positive claim?
Both. Both are asserting that compelling evidence can be found for their positions. And to assert that there is compelling evidence for something (i.e. to assert that something is "demonstrable") is to make a positive claim.