Shouldn't both sides have to justify their claims instead of one party having to do it?
Any claim - any claim, no matter how you phrase it or what its subject - is subect to a burden of proof. If I claim that the sky is blue, I have a burden to prove that claim. If you claim that the sky is blue and I claim that it isn't, we both have a burden.
I don't understand, because from a skeptic's point of view there's no absolute truth, so both parties should have the burden to prove their position and not only one.
'Absolute truth' is a red herring here. Regardless of whether or not such a thing as absolute truth exists, any claim still has a burden of proof. Unless one is a solipsist, absolute proof is not a requirement for acceptance of a claim.
But not all positions are claims about the subject.
Let's say that you claim that the sky is blue and I say that I don't believe you. The only claim I am making is that I don't believe, not that the sky is not blue, or that the sky is a different color than blue. I can support my claim to disbelief simply by referring to the only definitive authority that can possibly exist: myself. You can then conduct the trivial proof of your claim by pointing to the blue sky and if I am benig honest then my disbelief would be dispelled.
I don't understand this logic and it seems like a fallacy to me. Am I correct?
It depends on what you're specifically referring to.
If someone makes a claim and then says that they have no burden to prove that claim, then they are simply wrong. Their claim automatically attracts a burden of proof, by virtue of being a claim. This isn't necessarily an onerous thing, and they may be able to meet their burden with a trivial demonstration - such as pointing to a blue sky after claiming that the sky is blue.
If however the person does not make a claim about the subject, then that person does not have a burden of proof.
The confusion often arises because a person who holds a position such as "I don't know" or "I don't believe that claim" is not making a relevant claim, they are stating that they do not accept that the original claim is true. While this could be viewed as a claim about their internal state, that is hardly relevant.
A states "I flipped a coin and it landed on heads."
B says "I don't believe your claim."
A counters with "Prove that it landed on tails!" Regardless of whether or not
B does in fact believe that "it landed on tails," at this point
B has not made such a claim.
Note that this is not about the ontological truth of the statement, but about the belief states of the disputants. While it is true that the claim (the coin landed heads) is either true or not true, without some evidence to support the claim the respondant is not required to accept any particular truth value.
It is important to note that "I do not (believe/accept/understand/etc) your claim" is not the same as "your claim is false" or "a counter-claim is true." A statement of disbelief or non-acceptance of a claim does not automatically constitute either assigning a truth value to the original claim or assigning some truth value to a competing counter-claim, and it is an error to assume that it does so.
A classic example of this error is the many creationists who respond to "I don't believe you" with some variant of "prove that God does not exist" or "prove that the universe created itself" or some other such statement. This is a fallacious shifting of the burden of proof away from the claimant. The claimant has a burden to prove the claim, nobody else is required to disprove it, or to prove an alternate claim they did not make.
If on the other hand the response was "God doesn't exist" then the respondant has made a claim of their own, and thus acrues a burden to prove that claim.