As an actual communist what Marx meant by private property would be better called corporate property. Though the private/personal property division also works but your home would be considered personal property. And do note that this can get more complicated or differs depending on which branch of communism or anarchism youre specifically talking about.

Also to the anarchist anon, anarchism isnt against organizations, since people can organize into groups to get tasks done, and its specifically against vertical or non democratic heirarchy not all heirarchy. Unless you were talking about a specific branch of anarchism.

Communist-Anarchist Lesson today on HSE apparently! I am not too well informed in this topic so, I’m taking your word for it.