I had nose surgery for deviated septum and a hump. Something was also done to the nostrils as I have little scars on the sides. Too much cartilage was probably taken away and the tip dropped. Also some valve problem hence breathing is not very comfortable from that side. They said they will re-operate and put some cartilge from the septum to elevate the tip. I am not too keen on general anaesthetic.
Rhinoplasty is most commonly performed under a general anaesthetic. There is a possibility it could be performed under local but this really is dependent on the surgeon and the extent of the surgery needed. Different surgeons will have different views on this so you would really need to discuss this during your consultation.