A few key things a good PO must do:
- Don't change the Sprint goals mid-Sprint or change the Sprint length.
- Maintain the backlog. Have a good set of stories that fit in a Sprint (Ken Schwaber recommends six Sprints worth of stories) identified and prioritized.
- Insist on stories being "done" at the end of a Sprint. "Done" is a big topic on its own and a PO should clearly communicate what they consider to be done at Sprint planning.
- Participate in the review and planning sessions. Focus on the game shown at the review as well as the plan. Inspect and adapt.
We've found that "embedded POs" are best for our games. These are PO's that are part of a team, are available to give feedback during a Sprint and own the vision. That can answer to a higher authority of course (like a publisher or studio exec), but don't expect an exec to fulfill the key PO roles. The PO needs to be a bridge for many high level stake holders.
As with Scrum Masters, POs have access to great training classes that teach them how to be Product Owners.