How do your PMO and Product teams interact? We are in the process of building a product team to develop, launch and continuously improve a range of internal business products whose role is to closely interact with global internal business stakeholders, external vendors, and internal technical teams to deliver solutions that solve business problems. How have other companies governed these teams? (i.e.Who sets their budgets, resolves conflicts, connects product teams together, and ensures underlying data, technology, and security are in place? How do you report on progress and roadmaps to executive teams?) How do these teams interact with the PMO (project managers, change managers, business analysts)? 

In the end its about having a cadence. That depends on the way the PMO works. Depending on whether you are traditional Stage Gate or Agile your need for touch-points and the cadence may be different. In agile touchpoints are more frequent at times so you'll need a collaborative tool or system so that product managers can introduce new product initiatives and then once approved track the progress at a high level as PMO is providing it. 

But generally, product teams first capture the needs and create the draft business cases, apply for budgets and then send them to PMO for execution. So the first part which is drafting the business case requires heavy product management work in collaboration with the PMO, engineering and maybe finance, to get high level costing and time. When the project is launched then PMO gets busy and has to provide info back to product team to keep the product in-line with market and company needs and objectives. 

This is a complex matrix setup that needs careful considerations as at many points product management and project management/PMO might have conflicting interest. While product mind set tends to keep time (in terms of sprints/releases) and cost (in terms of team/resources) fixed and play to optimise scope while allowing varying scope, pure play project mgmt keeps scope and time as fixed while optimising cost.

This ofcourse creates conflicts when all three needs to be fixed by keeping both product and project mgmt working together.
The best way to look at it is keeping the essence of product mgmt (budget for the same team for the entire year assuming scope will require the effort needed to be put by the team), allow scope prioritisation at product manager level with stakeholders alignment, time critical features to be released while others to be fluid. Change mgmt should be handled mostly by PMO if possible with support from product mgr.

Do not go into feature wise planning and tracking at Gantt chart level mapping task and their delivery date etc. as it would be counter intuitive. Plan for releases and not sprints and keep team constant for reducing ramp up and down cost.