Hi chofontanilla,
To answer your question, anesthesia can cause side effects on the mother and baby. Epidural anesthesia, for example, can cause drop in blood pressure of the mom, nausea, shivering, headache, etc, and in rare cases, can cause respiratory depression on the baby, possibly difficulty latching when you want to breastfeed, etc. And another thing, when you do NSD with anesthesia, you may have more difficulty pushing because you can't feel the contractions or when to push (you actually don't feel much at all below your waist, which is why the doctor and nurses really need to teach you when to push).
But of course, not everyone experiences this, and most women who give birth with anesthesia don't experience problems at all. I actually wanted to give birth without anesthesia, but my OB discouraged it, so I had NSD with epidural, and didn't experience any problems. I still went through the labor pains (the anesth only inserted the epidural when I was at 8cm), then when I was fully dilated, I was wheeled into the DR/OR and gave birth with the help of the doctors/nurses chanting when it was time to push. I was wide awake during the procedure, and the baby came out perfect and was placed on my breast before being taken for the measurements/newborn care. Then came the placenta delivery, stitching, which I didn't feel (again, because of the anesthesia).
Afterwards, an hour or two stay at the recovery room where nainip na nga ako 'coz I wanted to go back to my room already, and I already had feeling below the waist. A nurse at CSMC brought my baby again and taught me how to get him to latch and breastfeed. Then finally, I was wheeled to my room, where I immediately asked for food kasi gutom na talaga ako. Nakakain and nakatayo naman ako kagad. And the baby was healthy... we also didn't have problems breastfeeding.
So, I'm glad that the NSD with epidural didn't pose any problems, and it was a really smooth birth all in all. It's really up to you and your OB what you decide on. But as with any medication, there's a possibility of side effects (which is why some women choose not to have it), though minimal lang naman.