While international oil prices have plunged this month, Saudi Arabia’s normally talkative oil minister Ali al-Naimi has been on vacation. Mr. al-Naimi’s absence from the fray—people familiar with his agenda say he was on vacation from the end of September and only returned to his office in recent days—is one symptom of the unusually high level of dissent within the secretive kingdom that has left it uncertain over how to respond to oil’s downturn. During past sharp oil price moves, traders watched for statements from high-level Saudi officials that might calm the markets—Saudi Arabia is by far the biggest oil producer within the Organization of the Petroleum Exporting Countries, pumping some 10 million barrels of crude a day. The 79-year-old Mr. al-Naimi, oil minister since 1995, is usually the kingdom’s key spokesman.