The definition of necessary is basic requirements for life or required to be done. According to the definition, I would have to say "No", Twitter in not a necessary.
As an educator we are always looking for new ways to engage our students and bring out the best in that student. Social media is such a part of our students lives, that I feel a student might show you how they understand a subject better using social media because they are so comfortable with it. So we need to embrace social media and find ways to bring it into our classroom.
As an educator, I would say "Yes", Twitter is a necessary!