There are many issues related to the role of Product Owner but I consider that sharing a Product Owner between several Scrum teams is the worst thing that you should face it.
Product Owner has to work full time with one Scrum team in order to participate in each ceremony such as Daily meeting (of course he would miss some but not more than once a week), Sprint Planning (without him the Scrum Master has to cancel and move to other date), Inception, Release Planning and so on. In addition, Product Owner is the bridge between stakeholders and team members.
Part-time Product Owners
If you have to deal with Product Owners that have extra duties you would be in trouble. Why? Because the Product Owner would not be focused on the project and he won’t be available for team members so your team members will be blocked if they need any feedback, answer or decision from him.
Sharing one Product Owner
Imagine you have a Product Owner that has to attend two daily meetings, Sprint Planning, Sprint Review (or demo) and Release Planning. Well, this is a totally madness because he has to write user stories, refine two backlogs, prioritize Sprint backlog and prioritize Features as well.
The logical conclusion of this: most of the time the Product Owner has not feed any of the backlogs and team members will get angry about that.
What about the stakeholders?
Well, if the Product Owner does not work full time how can he deal with different stakeholders? In fact, Product Owner should meet every week with stakeholders to notify them the latest news and of course to convince them to attend the Sprint Review or demo in order to get honest feedback for them.
If your stakeholders are not concerned about the project, the Product Owner has to analyze the impact and the consequences of their lack of interest. My suggestion is to get another stakeholders who care more about the project or assume the risk that the project will be useless. It means nobody will use in Production.