I think early teens if they are comfortable with being alone. I think my son was either 12 or 13 when he started having a physical without me being there. But..I did give the doctor some things I wanted him to talk to him about such as STD, bike helmets, dangers of binge drinking, that sort of thing. Of course, I also talked to him about it but figured the reinforcement from his doctor couldn't hurt

I promise I'm not a helicopter parent, honest! 
The doctor asked him what kinds of things he liked to eat & he literally answered "I don't know" (he was much more interested in the fake hand/foot/spine display the doctor had than answering questions). 