Despite their differing priorities a Product Owner (PO) and Scrum Master (SM) must be on the same team. More specifically, in a successful scrum team the PO must be a partner to the SM. Their adversity comes from the roles they play. The sentiment applies equally well to Agile Project Managers and Agile Coaches. That said, framing it in the context of Scrum makes the narrative easier.
The PO is beholden to the customer or end-user as well as the business. Their main function is to maximize value delivery. The SM is beholden to the team. Their main function is ensuring and enacting scrum through servant-leadership of the team. With no SM a PO might very well burn a team out by requiring too much of them. Without a PM a SM might hinder value delivery by shielding the team from more than they should or allowing more experimentation than the business can support. Hence a partnership must form between the PO and SM for a successful team to emerge. The adversity arises due to the difference in priorities between these two roles.
The PO and SM have to work together to find the right balance between pushing the team for higher performance and sustaining the team for long-term endurance. Together they need to know what level of change is tolerable by the organization, the user, and the team at any given point. Using this knowledge, they need to decide how much time the team can spend on self-improvement versus value delivery every iteration. In a solid partnership they might be able to discover ways to accomplish both improvement and value delivery at the same time.
Just as changing around a development team causes a reset of the teams development changing the PO or SM does the same for their partnership. Having multiple PO’s on a single product exacerbates the problem, and is a bad practice to begin with. The PO is supposed to be the one and only person who can speak to what the most important deliverable is at any time. If there are multiple people trying to fill the role there will inevitably be a time where they disagree, potentially causing confusion when the SM is looking for clarification or guidance. Similarly, a dedicated SM can establish a relationship with the PO and continue through storming and norming to performing. Choosing to instead rotate the SM role among team members can result in that relationship never progressing beyond the storming phase.
When a PO and SM are on the same page it becomes easier for the team they are on to deliver the best value possible. If they are at odds then their conflict itself will become a value blocker for the team. Keep both roles dedicated, on the team, and in a partnership for the best outcome possible.