I would take a step further and ask: why are you doing retrospectives in the first place?
The answer most likely is: because we want to learn (from our past experience)
And now we're coming to the point: who should feel accountable for learning?
The answer: everyone.
Lessons learned, postmortems and retrospectives produce specific feedback, because usually it is very specific (project-related) and can't be easily applied in general. But then it is also feedback which is potentially applicable not only on PM level but also on organization, team and individual levels.
So the basic answer for the question who should feel responsible for making use of postmortems is: everyone in these areas where it does make sense. PM should use it in project management activities, functional managers on team levels and individual contributors on their level as well. Stakeholders, especially those high in pecking order, should take care of conclusions regarding organization as a whole.
It is a cultural thing -- usually you're either a part of learning organization and such behaviors are rather common or your company does postmortems only because it is a part of their project management methodology and hardly make any use out of them.
It is even more important in organizations where project teams are changing over time, and this is my impression on your company basing on the question. If PM is now in this project with these people and tomorrow works on another project with completely different people they just can't be the only role responsible for applying lessons learned. It just won't work that way.