It all comes down to how you feel. As others have told you, discuss this matter with your anesthesiologist and the surgeon that will perform the operation. If you're extremely scared to be conscious, then you have every right to be asleep during the procedure (though I'm not sure how it works in the U.S, I guess it depends on your insurance?).
I myself have been through five open-heart surgeries due to a complex congenital heart disease, plus three angioplasties under general anaesthesia. I didn't wake up during any of those. I've also had several painful procedures done under local anaesthesia, a few which were so painful that not even the sedatives they gave me to calm me down worked. I was in full panic, but couldn't do anything about it but try to scream.
One interesting thing is that they always give you sedatives and other things to "make you forget" before you go under general anaesthesia. They've never worked on me! I've always remembered everything, up to the point when they sedated me. When I was little, the sedatives actually made me even more hyperactive... Back then, they didn't know about my AS, so they attributed this to the fact that I was born a red-head. But now it all makes sense!
I also had to get my head stitched up twice when I was two and three, plus my lip stitched when I was four, because I fell a lot (clumsiness). This was done under local anaesthesia. I don't remember that it hurt, but then again, I was so little that it's hard to remember.