Whenever the narrator produces loot tables for the characters, there may be some debate as to who gets to take the Loot action. As stated, any table can only be rolled on once (by one player) - surely the narrator can provide a table for everyone, but typically the loot will be more restricted than that.
So that begs the question of which player gets to take the action.
From a meta-gaming perspective, there likely will always be a right answer here. That is, the player with the highest luck (or other modifiers) should produce the best results for the group. The problem lies in "fun" potential; rolling for loot is fun - loot tables are specifically designed to give players an entertaining consequence-free gambling feel. "What did I find?" is something that most players of tabletop roleplaying games want to ask at some point.
It is ultimately up to the group and the narrator how best to handle this approach. Some groups may feel that their best plan is to let the top looter do her work, and simply share the spoils. Other groups may prefer the "he who found it first gets to loot" approach.
If the group is having a hard time with this, or there appear to be some hurt feelings, the narrator may wish to implement a round-robin approach, where each player gets a turn to roll loot (the narrator can track this on the story sheet, if desired). One drawback to this approach that the "best looter" approach does not have is if a character has specifically selected a species or tale that enhances their looting capability, they may feel like this investment was wasted if they don't get the opportunity to exercise it. In this case, it may be worth the narrator noting that that rolling the loot action isn't fundamentally different from rolling to attack, or rolling to charm - if a character is good at that task, the group should consider letting them be the ones to do it.