I don't know how many print producers still exist out there in agency land, however, on the oft chance they still exist (most have morphed into integrated producers or left the business) here is a typical print production manager job description.

Production Manager
Supervise assigned client's print production projects.

Responsibilities:

Coordinates production activity between project, account and creative groups during job development.
Maintains contact with various printing vendors (Sheet, Digital, Web, Form, In-Line, Promotional Material, Dimensional)
Maintains detailed project records, from spec sheets to estimates to purchase orders.
Supplies specifications for vendor estimates.
Reconcile all assigned project billing.
Reviews and supervises all production materials generated from their assigned projects.
Acts as advisor for the creative and account members of the team during creative development to meet the budgetary requirements of the production process.
Maintains ownership of all projects in their group.
Issues POs on a timely basis.
Checks all vendor invoices against both POs and estimates before signing off. Signs off on bills in a timely basis.
Maintains and keeps up-to-date on all aspects for printing, production and postal requirements.
Advise their team of the best formats and efficiencies.
Gives creative solutions.
Provides quotes to Account Staff/Clients.
Requesting quotes from suppliers.
Negotiating prices with suppliers.
Scheduling of print jobs between suppliers.
Order any production related materials.
Press approvals.
Help coordinate with the studio, traffic managers, and suppliers to ensure all deadlines are met.
Proofs incoming materials for conformance to specifications and monitors delivery schedules; maintains control of projects.
Maintains an adequate proofs and collateral file system.
Adheres to the agency's operational and financial policies and practices.
To develop a close working relationship with the Creative Department conferring with Art Directors so as to obtain artwork suitable for good reproduction, iron out any problems, and receive specific instructions.
To suggest to Art Directors revisions in finished art that might facilitate production or reduce its costs, or improve final reproduction in specific media.