Welllllll.... don't speak too soon about humans not eating their placentas...
It makes sense that the other mammals eat their placentas - they are quick source of protein for a ready made meal, especially in those critters that den up with their newborns for a period of time without eating. There is oxytocin in there that helps the uterus continue to contract and in the case of cats, encourage the rest of the kittens to be born. Other hormones in the placenta help boost milk production.
Finally, in herding grazing animals, eating the placenta helps hide evidence and the smell of the birth from predators who would like nothing more than newborn antelope for a snack.
When you look at it logically, you kinda wonder why we don't eat ours... Still it is not something that most humans are interested in - the 'ick' factor.
Well, not to gross you all out, but I teach natural childbirth classes, and we get all kinds...
There are actually recipies out there for preparations made from placenta - from soup to roast. In China it's considered a traditional medicine (dried and powdered) and taken for female complaints, which makes sense with the high concentration of hormones such as prostaglandins and oxytocin in the tissues.
Occasionally I get a couple who is interested in preparing their placentas. I personally know one lady who tried making stew out of theirs - said it really wasn't too tasty...
Lots of people do keep their placenta for various ritual or symbolism. Many plant theirs under the roots of a favorite tree, or bury it and then plant a tree in honor of the new baby. (Good fertilizer...) Others make 'placenta prints' - think sponge painting.
Funniest placenta story was a couple who did a hotel birth with a midwife (midwife couldn't come to their home because was in a different state where she wasn't licensed to practice) Anyway, they delivered the baby, all went well and they checked out of the hotel about midnight and went home. Just as the mom was getting settled down with the baby to bed, she sat upright and said "Oh honey, we left the placenta!'. They had left it in the mini bar refrigerator. So, off the dad goes in the middle of the night to retrieve the placenta. Just said 'uh, we forgot something' to the clerk. Can you imagine what the cleaning staff would have thought if they'd found that in the fridge??
It takes all kinds in this world. and the older I get the more I learn about people, the less judgemental I've become about their belief systems...