As a preliminary consider what the two "burdens" are according to argumentation theory and how they may be abused.
According to Wikipedia, the burden of proof is one of the key components of argumentation theory.
Establishing the "burden of proof" – determining who made the initial claim and is thus responsible for providing evidence why his/her position merits acceptance.
This "burden" belongs to the one making the claim. Once that argument has been presented another key of argumentation theory is the "burden of rejoinder":
In a debate, fulfillment of the burden of proof creates a burden of rejoinder. One must try to identify faulty reasoning in the opponent's argument, to attack the reasons/premises of the argument, to provide counterexamples if possible, to identify any fallacies, and to show why a valid conclusion cannot be derived from the reasons provided for his/her argument.
One way of viewing these two burdens is that they set the ground rules for which side speaks first while the other listens until they get a turn.
As politicians and advertisers are aware repetition is important in persuasion:
Psychology has long studied the non-logical aspects of argumentation. For example, studies have shown that simple repetition of an idea is often a more effective method of argumentation than appeals to reason. Propaganda often utilizes repetition.
Since there is a burden for both sides in an argument and since repetition is important to persuasion, when one side of an argument refuses to present their case, even to simply repeat it, this should make one wonder why they are not taking advantage of their opportunity to speak?
Rational Wiki provides examples of ways burden of proof is abused such as shifting the burden of proof:
Fallacious shifting of the burden of proof occurs if someone makes a claim that needs justification, then demands that the opponent justify the opposite of the claim. The opponent has no such burden until evidence is presented for the claim.
Other ways to abuse the orderly taking of turns with burden of proof is when there is no way to falsify a claim such as simulation theories or exceptional clams such as miraculous cures or both theist and anti-theist claims for or against God.
With that preliminary let's consider the scenario presented by the OP:
Say person 1 believes x because a valid authority said so. Person 2 is skeptical of said claim and asks person 1 to present the evidence that supports the claim. Person 1 says he doesn't have to and it's on person 2 to provide evidence the valid authority is wrong.
For clarification person 2 agrees the authority is valid and is aware that the valid authority has made this claim.
Person 1 made a claim and so has the burden of proof. Person 1 presents evidence from an expert witness, the "valid authority", that person 2 acknowledges as an expert.
Person 1 is done presenting evidence. It is now person 2's turn to argue against the evidence, the expert testimony, by identifying faulty reasoning, attacking the reasons/premises of the argument, providing counterexamples if possible, identifying any fallacies, showing why a valid conclusion cannot be derived from the reasons provided, to paraphrase Wikipedia.
If person 2 does not wish to assume the burden of rejoinder this should raise suspicion in the minds of those listening to this argument about the value of person 2's side of the argument.
Rational Wiki, "Burden of proof" https://rationalwiki.org/wiki/Burden_of_proof
Wikipedia, "Agurmentation theory" https://en.wikipedia.org/wiki/Argumentation_theory