Many thanks to the reader who emailed me to mention the KSA MOH's Statistics Book for 2009. He cites numbers on p. 107:
• The total number of physicians (including dentists) is 25,832.• The total number of Saudi physicians (including dentists) is 5,831 (22.6% of the total)
He also notes that roughly half of the nurses in the Kingdom are Saudis—and the other half, of course, are "residents" or expatriates.
More recently, Saudi Gazette reports today that a "recruitment mafia" is getting doctors with fake credentials into the Kingdom, and notes that when these doctors are found out, the MOH response is not to deport them or jail them, but to assign them to "far away places and remote areas."
Update: Thanks to the commenter who noted this ArabianBusiness.com report, pointing out the political nature of employment in the Kingdom and the Saudi heathcare graduates trying to find work with better job security.