Glad to see this thread up and running again. I was just thinking about it.
Okay, so this is a GREAT BIG AYLR (ask your local rabbi) question, because it will depend on your specific situation.
On the surface, it seems like it would be a simple issue...eating something from a non kosher animal (human) is forbidden...except in the case of honey. There might be another thing that I am missing here. But breastmilk, obviously, is THE PREFERRED food in Judaism in MOST cases, and this is also from a "non kosher animal" (even though most religious authorities hold that we are not "animals") so immediately you would see that this is more complex and not so simple.
I have heard of several rulings that have permitted women to eat their own placentas as an issue of pikuat nefesh (saving a life), because of it's proven properties to combat post-pardum depression. Of course, your local rabbi would need to be educated about this usage, so you would need to either find an expert in the field or find one who is willing to read some of the reseach that you and your practitioners complile for him on the subject. In this case, it is preserving the life/health of the mother and by extention the baby as well so it has been permitted.
I've heard of it being not permitted because it is like eating an animal while it is still alive, which is forbidden.
I've heard of the blood issue being averted because A. it's not from a kosher animal anyways, so the rules are different and B. you can prepare it like liver (i.e. chicken liver) in a kosher way which would get rid of the blood issue.
I've heard of it being allowed so long as there is not a "taste" (like swallowing pills, if it's not a "food" then it isn't considered a food and rather a trufah (remedy/medicine) and so if you just swallow it you can take it. This ruling has been used for other things, like animal thyroid from non kosher animals, as well.)
So, it would depend on your particular situation and your particular reason for eating it and your own personal rabbi would need to make the ruling.
Also interesting to note is that the Jewish custom for organs or other human body parts is to bury them (that is the halacha). Some rabbis have ruled that this applies to the placenta as well (though many have not gone so far to make this ruling) but that is something else to think about, as well. Aparently there was a custom in some Jewish communities of burying the placenta and planting a tree above it...that tree would grow and become part of the chuppah (wedding canopy...the poles that hold it, for example) for the son or daughter for whom the tree was planted. A nice custom, dontcha think?
I, personally, bury mine. I have burried each placenta in the yard where we were living at the time (actually my dh did the burrying), except in the case of my miscarriage, where everything was too small to sort out and it all, basically, went down the toilet.
I intend to bury the forthcoming placenta in our yard here, as well. Perhaps under an already planted tree (we had two planted when we moved in). It makes me feel like we are tied to each place we lived in with the kids...even though we are STILL renting and have had to move (and will again).