If i understand your description right, your scrum dev team consists of 2 parts: dev team, and qe team, and each has a manager, who is the ultimate decision maker for their respective teams, and who also has right to assign tasks to their team outside regular Scrum process (backlog, etc). Am i getting it right?
For the task assignment outside of Scrum, it should be avoided if possible. Talk with the QA manager and see if it's possible to follow the regular rules (new tasks should be added to the backlog, go through grooming, planning, and be accepted by the team at the right moment).
There may be the case when it is not possible (high priority tasks that need immediate attention). In this case, it is best to track them outside of Scrum, so team's velocity will be decreased because they need to do outside work. This situation, of course, is an impediment, but it can go to impediment log, and retrospectives can be used to find a way to solve it.
For the other part, if the tasks are not evaluated by the team itself, but there are figures of authority who make decisions for their people, it is well against Scrum rules, but it happens most of the time. It reduces team commitment, and maybe precision of estimates too. In order to decrease those harmful effects, you can reinforce the following two things:
- Managers are empowered to make commitments instead of their teams, but then, they are solely responsible for keeping them. So you can replace the democratic team model with a hierarchical one, but they get the whole package, not only the benefits :) However, please be aware that this stretches the methodology so much that it will not be regular Scrum any more.
- You should help managers implement democratic team model at least on their subteam level (discuss decisions before making them, collect input from the team members, etc.)