Even within departments, TAs may have different levels of responsibility. For example, some TAs are assistants to professors of record. Course assistants sometimes lead sections, grade papers and tests, oversee labs, etc. Other assistantships involve the TA serving as an autonomous instructor in the classroom. Even in situations where the TA is the instructor, responsibilities widely vary.
Some departments provide a syllabus and determine the course objectives.
Other departments expect the TA to set grading polices, write a syllabus, and to construct tests and assignments. There are many variations in-between. Teaching One's Own Section. The TA is assigned to teach his or her own class. The amount of responsibility will vary from department to department.