This is dependant on the surgeon choice. Local anaesthesia with or without sedation is commonly used. The main advantage of operation under local anaesthesia is that the eyelid & eyeball movements can be checked at time to surgery to ensure adequate correction and also no excessive removal of the eyelid skin.
When eye bag surgery is combined with other facial surgery such as upper blepharoplasty, facelift or nasal augmentation or chin augmentation then some surgeons may prefer general anaesthesia for the rejuvenation of the entire face.