Maturity. We try to introduce the child to giving his own shots around 8 years-old, though it really depends on maturity level. No matter when a child takes over giving himself insulin, parental supervision is always necessary.
8-9 years old. A child with diabetes needs some control over their new lifestyle. At age 4-5 years old, allow the child to "poke his own finger" for glucose check. By age 8-9 years old, teach the child to give his/her own injections, after parent has drawn up the syringe. By 11 to 12 years old, allow child to draw Insulin syringe with parent watching. By 13 to 14 years old, he can give his own injections.
When able. When he/she has demonstrated skill to administer shots and the responsibility over his/her disease.