Hello WWYOnly a small, limited tummy tuck, probably without muscle tightening would be done with a local anesthetic. Most patients require a long procedure that would be uncomfortable to be done well without a general anesthetic. Though for patients a general anesthetic can be a scary thing to consider, especially for cosmetic surgery, it is very safe and you are being looked after by an anesthetist constantly while you are asleep.
Yes it is possible to do a tummy tuck under local but it depends on your preoperative findings and your personality. Discuss this with your doctor.