What is the jurisprudence of Islamic common law about using placenta products for whitening by women to beautify themselves?

Praise be to Allah, the Lord of the Worlds; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.

If the “whitener” is taken from placenta of live animals, it is forbidden as the Prophet (Sallallahu Alaihi wa Sallam) said: "Whatever was cut from the animal while alive is considered Maytatah (forbidden)". [Al Darami]

It is known that the Meytah of animals is forbidden to us Allah Says (interpretation of meaning): {Say (O Muhammad SAW): "I find not in that which has been inspired to me anything forbidden to be eaten by one who wishes to eat it, unless it be Maytatah (a dead animal) or blood poured forth (by slaughtering or the like), or the flesh of swine (pork, etc.) for that surely is impure, or impious (unlawful) meat (of an animal) which is slaughtered as a sacrifice for others than Allâh (or has been slaughtered for idols, etc., or on which Allâh's Name has not been mentioned while slaughtering). But whosoever is forced by necessity without wilful disobedience, nor transgressing due limits, (for him) certainly, your Lord is Oft­-Forgiving, Most Merciful."}[6:145].

Now, if the “whitener” was taken from the placenta of a woman, then it is prohibited to use it.  Indeed, using human being’s placenta in such products is a form of humiliation of the human being hence in contradiction to the veneration that Allah has given to human beings.  Allah Says (interpretation of meaning): {And indeed We have honoured the Children of Adam, and We have carried them on land and sea, and have provided them with At-Taiyibât (lawful good things), and have preferred them above many of those whom We have created with a marked preference.}[17:70].

Allah knows best.