Can the customer be a SCRUM Product Owner in a project?

Question

I just had a discussion with a colleague about the Product Owner role: In a project where a customer organization has brought in a sofware developing organization (supplier), can the role of Product Owner be successfully held by the customer organization, or should it always be held by the supplier?

I always imagined, that the PO was the supplier organizations guy. The guy that ensured that the customer is happy, and continously fed with new and high-businessvalue functionality, but still an integral part of the developer organization. However, maybe I have viewed the PO role too much like the waterfall project manager.

My colleague made me think: If the customer organization is mature and proffessional enough, why not let a person from their camp prioritize the backlog?? That would put the PO role much closer to the business, thus being (in theory) better to assess the business value of backlog items. To me, that is an intriguing thought. But what are the implication of such a setup???

I look forward to your input.