Activity completion info can be viewed by managers, teachers and non-editing teachers (and any other users with the capability report/progress:view) from Course navigation > More and selecting Reports > Activity completion.
Teachers can mark activities complete on behalf of students by clicking into the relevant completion boxes. This requires the capability Override activity completion status which is enabled for editing and non-editing teachers by default. Activities marked complete by the teacher display with a red border in the activity completion report.
Teachers can also mark activities as not complete with the exception of quizzes which are automatically marked. Activities marked as not complete by the teacher also display with a red border in the activity completion report.