Yes, it is a full-time job, where the PRODUCT owner works with the team sprint by sprint for effective delivery of the COMMITTED work. In the case of Release Product Owner, it again requires full-time INVOLVEMENT where they CONNECT with the customers to understand their need and set expectations on the milestones and delivery. They work on managing the dependencies on the product which might extend to different teams, this requires a lot of effort because if any of the dependency goes unnoticed it impacts the overall release plan. 

They also work to keep the backlog stay up-to-date which ordered and prioritized requirements, this is a critical aspect as the CLIENT satisfaction is the key, they need to stay competitive in the market. Underestimating the role of the product owner or the release product owner can give a big set back to the teams and the organization, they should be given space to try out things and have them increase the collaboration with the customers and the teams.