It depends on the reason for the episiotomy hun. If it's because bubs is slipping back rather than crowning the do it at the height of a contraction while the head is bulging forward and the skin is stretched tight, ditto the episiotomy done to prevent a tear. But if you have to have one due to needing forceps the head isn't usually on the perineum, so they will numb you first if they have time. I know one women who had the all-too-normal failure-to-progress/augmentation/foetal distress and they wanted to section her, saying her son was obviously too big to come down, and she refused, saying she wanted to delivery vaginally, and the ob said, ok, well then i need to use forceps. He got them out and then said "because baby is so distressed, i need to make a small cut to get him out faster, there's no time for an injection, this might sting a little" and then cut a mediolateral episiotomy so deep she had to have her left glute stitched by another surgeon before the Ob could repair her pelvic floor. She said it was agony, far worse than the labour or birth, and she has PTSD. Her son was under 6lbs.

So basically it comes down to how much time you have, the reason for the episiotomy, and the compassion (or lack thereof) of the care-provider whether they numb you or not. I would imagine in the birth centre there are very few episiotomies, and those which are done are done carefully and compassionately, which is possibly why they use anaesthetic.

Bx