I don't think it really matters. In fact, you don't even have to lie down at all.
Dr. Clark stresses that you lie down immediately after consuming the olive oil mixture. She says to lie flat on your back with your head raised high on your pillow.
But, I have done it all ways and I can tell you that which side doesn't really matter.
Use extra virgin olive oil. It's better. But, if you're in a pinch, regular olive oil will work, too.