I have attended three shoulder dystocias now. One as a doula in the hospital, the other two as an apprentice at home. One was just a little over a week ago. The first homebirth SD I attended was predicted because the mom had two previous SDs, so the team I was working with went through a full rehearsal of all of the maneuvers that we were going to use and felt prepared. It helped a lot because it can be a pretty panicky situation.
With CPD, the head won't descend into the pelvis. This is a lot less dangerous than SD, because essentially the labor just stops progressing. You can make a decision to do a c-section, then take your sweet time setting it up and the baby will be just fine in most cases. With SD, once the head is out, it is a matter of minutes before the baby is brain damaged or dead. When babies have SD, their heads will often turn purple-black because of a buildup of veinous blood. The babies will often also have a very squished looking face because the chin is outside the perineum and the shoulder trapped behind the bone within. Babies can get nerve damage because the nerves in the shoulder get stretched or torn, causing a kind of palsy. They can also suffer broken bones because of the force used to birth them, either on the mom's part or, more likely, on the care provider's part.
SD is something that scares the life out of a lot of providers because action needs to be taken very quickly and the potential for injury to mom and baby is high. In one study I read recently, the rate of injury to baby was over 30% in the hospital setting. Fortunately, with the three I have attended, the babies have all been fine. One was born white and floppy with a black head and needed a full resuscitation, the others were born with good apgars, but all were in good shape at 5 min. There were postpartum hemmorhages in all three cases, but they were resolved successfully.
There is a maneuver, called the Zavanelli maneuver that involves shoving the baby back up into the uterus and performing a c-section, but the injury/death rate is so high for it that most obs won't even try it. So, in most settings, that is not an option.
The other maneuvers center around two plans of action, or a combination of both -- change the shape/size of the pelvic outlet and change the position/presentation/shape of the baby. Deep lunges, rotating mom to her side or hands and knees, and the McRoberts maneuver, where the mom's ankles are pushed up by her shoulders, are all things that open up the pelvis in one place or another.
With the McRoberts position, it is possible for a provider to use super-pubic pressure, which is essentially an effort to push on the lower abdomen where the baby's anterior shoulder is so that it will pop under the pubic bone.
Almost all of the other maneuvers that deal with the baby involve the provider reaching hands up inside the mom to try and change the baby's position/presentation/shape. This is where the episiotomy comes in. When a mom is on her back, it can be pretty hard to reach in with one or two hands to where the baby is and feel for impacted arms/shoulders, to get a good grip to turn the baby, etc. So, many providers will cut large episiotomies to provide room to reach in on the sides because the pelvis is in the way in the front. In some OB textbooks, cutting the epis is the first step, so if a provider has learned this way, he/she might think of it as being a necessary part of the procedure, no matter what position the mom is in. On hands and knees, it is more possible to reach in and access the baby (because the bones aren't in the way), so epis are often not necessary in that position. In the two homebirths I have attended, no epis were cut, moms were in lunging or hand & knees positions, and they did not tear. In the hospital birth, they used McRoberts and super-pubic pressure and also did not need to cut an epis. I believe there was a 2nd degree tear there.
The other reason why a provider might cut an epis is that he/she wants the baby to not encounter any resistance once it clears the pelvis because he/she is worried about how long the birth has been delayed already. In at least two of the SDs I attended, the baby's chest was considerably larger than its head, so I suppose that this could be a legitimate concern -- essentially that the baby could take a little time passing over the perineum after the shoulders have released.
Cutting an epis will not free a shoulder that is still stuck under the bone, though, and I have heard more than one birth story where there was a SD and the provider just kept cutting and cutting, hoping that the baby will just pop out, without doing anything to change the position of mom or baby. I think that sometimes people just panic and lose their heads.
Hope this helps. Sorry it is so long!