Facebook needs the right to censor to protect people from, at the very least, the most hurtful and provocative speech and images. Without some form of censorship, it will erode into a path of hate speech, porn, and advertising with the customer base turned off and leaving. Being publicly traded, they will always walk that fine line.
Facebook should have all the rights in deciding what should be censored. It has provided people with a social platform to communicate and therefore just as we have laws to govern us in the real world, measures should be in place to control and regulate what goes on on that platform.
Facebook should have the right to censor because it is a privately owned company. If someone owns a company, no one should be able to dictate to that individual about what they can or cannot do as long as they are within legal bounds. The public has a choice of whether or not to use the company's service or not. Other platforms are available that the public can use to communicate with others or post photos, videos, etc.
Yes, Facebook should have the right to censor content. It is a privately owned company and has the right to decide which content is allowable on the website. Any content that they deem to be inappropriate for their website should be removable if they deem it necessary to do so.