At my current company (automotive stamping) quality department does the generation of : PFD,
PFMEA & PCP (Competency as mentioned by Jonny)
My previous company (window regulator) the manufacturing is responsible for creating and maintaining (living document) of the PFD & PFMEA. While the quality engineer is responsible for using the PFD, PFMEA to create and maintain the Pre Launch & Production control plan.