It’s entirely unclear, as is much of the Westerosi legal system. We know from WOIAF that Jaehaerys I and Septon Barth wrote a unitary legal code for Westeros, but we don’t know how it’s enforced, especially because lords still have the right of pit and gallows.
So here’s my best guess: maesters in the Citadel are educated in the law from Jaehaerys’ code and the various precedents of interpretation. They advise the lord as to what the law is (and depending on the lord, they might be the judge in all but name) but don’t have the power to make the lord follow it. (Although, as the keepers of the ravens, they do have the power to potentially inform on their lord…) However, a decision that violates the code might well be reversed by that lord’s liege lord, because it’s technically treason and would (if found out about) bring the wrath of the Iron Throne down upon the lord might their liege who failed to uphold the king’s laws.
On the other hand, that all depends on whether anyone with influence gets the word out. If a lord decides to hang a common peasant who the law says shouldn’t be hung, the chances of a successful appeal or reproach to the liege lord are extremely slim, unless some other knight or lord makes it an issue. (For example, if a lord hangs another knight or lord’s peasant (like Dake from Sworn Sword), or the peasant in question happens to be a servant of someone important who takes offense) A merchant or very rich peasant might have enough money to pay a maester to draft a formal complaint or to bribe the liege lord into hearing their case, so that’s riskier. A knight or lord, and now you may very well be talking about either an appeal or a rebellion.
If my guess is right, the Master of Laws might well get reports from maesters about legal decisions made out in the provinces, especially if there was a significant legal dispute between lords, to ensure that the King’s laws are being enforced. It might even be the case that (similar to how the Hand of the King can issue judicial opinions in the absence of the king) the Master of Laws acts as an intermediate court of appeals, prior to the King or Hand getting involved.
But honestly, this is something that GRRM needs to clear up, which is why it’s on my list of questions to ask him if I ever meet him.