Product owners are great to have within a team. They spend time understanding the needs of the product from a client side and disseminate that information to the development teams. With that knowledge, should they be involved in the testing process? I would say yes. In fact I find that they have to be ingrained in the process from the start to have good visibility of what is going on.
Static testing is one of the key things they would need to be cognizant of. They need to have the assurance of the information that they are passing of is detailed enough to get the work done. Now in an Agile environment that level of detail may not be there from the start to get the exact product change they are looking for, yet they should have enough information to allow the team to see the path that they must go down.
Additional understandings of the process, such as issues found, changes to stories and subsequent impacts help the overall process. Clear paths of bidirectional communication allows for effective risk analysis and decision making to ensure the product is what is needed by the clients.
Now onto dynamic testing. Why shouldn’t the Product Owner be involved in doing some testing of the product? In the end they should have some idea of what the product looks like and how it processes the information according to client needs. Now does that mean they test each sprint or during the beginning stages of the test cycle? I don’t think so. There are potentials to find too many issues that could raise unwarranted alarms. Let the people running the tests get those out of the way and come back to the product owners for clarification or advice on what was found. In the end they should be a part of some form of UAT type of cycle prior to completion. Let them get their hands a little dirty using all the information they gathered throughout the process to see if it is what they want.
In the end it is their product and they should have an idea of what it does and how the clients will feel when they start using it after it goes into production. They may have an unusual workflow that they found out a client does and will need to see if it is run through. Whether it is the test team themselves or they try it to find potential issues. This level of understanding along with strong communications will help with up front quality improvement along with closing any gaps with testing.