I would say, whoever gets in there first with the Other Duty has precedence. If another player subsequently rolls an OD that contradicts the one already appointed, he may not select that player; if there are no other options, re-roll.
eg. If a Minbari player rolls Vorlon Directive, then a Drakh player rolls Whispers From The Shadows, the Drakh player may not select the Minbari player. If it goes the other way, the Drakh player rolls WFTS and selects the Minbari player and then the Minbari player rolls Vorlon Directive, he should re-roll since his attack is already determined.