For weed, I say 14 or 15. I started at 16, and I realized that I was kind of a bad, egotistical, overstressed person before I began. Just weed can be very powerful for a youngster with no prior drug experience.
For DMT, it's difficult to say. I began at 17/18 and am fine.
I'd say that if you're looking at it through a moral lens, then it really depends on the person's maturity level, and maybe not even that. Psychedelics forced me to grow up; I would be much younger mentally now without them. So an individual would have to be gauged to see if they'd actually benefit from use, or just be catapulted into insanity (for having to face their true selves, for instance).
Through a physical lens, you would have to ask a researcher/doctor. Of course when they say teens shouldn't smoke weed, I feel that's mostly propaganda, but 13 year olds should probably not be eating LSD blotters. I wouldn't be so quick to tell others what to do or how to live their lives, but having all of the information first is helpful to everyone.