Say someone films a criminal or even a police officer engaged in an act of violence. Yes, they should legally be allowed to post those images. The public has a right to know.
Of course some websites may decide not to allow these things to be posted, and that is their right as private organizations. But news outlets, whether professional or amateur have a right to be free from government censorship in reporting events.
If anything it is better that these things get posted online. Then the proper authorities will learn about what happened and take the appropriate actions to bring the perpetrators to justice, and in the case that the perpetrator of violence is one of the authorities the public will demand that his conduct be investigated.
Posting violent images can be offensive, but offensive expression is protected. Many images are hurtful and not appropriate for younger viewers, but I'm not sure they should all be censored. Clearly hateful images targeted at a specific gender or ethnic group should not be covered by freedom of speech, but violence alone shouldn't be enough to be censored.
While I cringe at the idea that people might post violent images of leaders or ethnic groups getting assassinated or murdered, you can't ban all "violent" images. Sometimes, people don't know what cops do to black victims until it blows up on social media. How would we sort out those sort of 'violent' images from ones suggesting actions? It's too subjective.
No, freedom of speech does not sanction the posting of violent images. You can have freedom of speech as long as you are not hurting someone. For example, you can't bully and insult someone and then hide behind "freedom of speech" to defend your actions. Similarly, you can't post disturbing violent images and hide behind that freedom. As with every freedom, there are guidelines and common sense limits.
Freedom of speech does not mean freedom to be vile or freedom to show violent or distasteful images. As adults we should be able to express ourselves openly without making someone cringe in the process. In this day of social media everyone try to take the easy way out and try to get their point across with shock and awe instead of using intellect. There is already too much violence in our everyday lives and we need to do better for our children, the future of this world.