Why are you assuming that the burden of proof is on the person without theistic beliefs?
Agnostics would say "I cannot find a proof either way, so I do not know" and as such need show no proof as they make no factual claims, they only state their uncertainty on the matter.
Theists would say "I believe God exists" and as such may be called upon to try and show a proof for their belief
Atheists would say "There is no God" and as this is a definitive claim, there must be a proof for it to be considered a true fact.
If you make a claim that something is a certain way(such as God does or does not exist), you may be called upon to provide a proof of that.
If you claim that something cannot be known(such as an agnostic claiming it is not possible to know if God exists), you may be asked to provide a proof of that impossibility.
If you merely say "I have yet to see a convincing proof" then you have nothing to prove as you are stating an opinion about your own experiences and not attempting to assert anything as a fact.
Facts can be proved or disproved and any statement of fact can be challenged as such.
Does that help clarify things?