Don’t use a “page”, in the meaning of a WordPress (static) page content type. Use either comments or let user register and get the either the contributor role or the author role.
Comments are easy. Allow comments generally and allow comments on the specific post (or page, if you must, actually).
A contributor may not publish, just write and save as draft or pending. There are plugins that will let you receive an email whenever a post is pending. An editor will then publish it. An author may publish directly, but not touch others users content.
To limit which category contributors or authors may use, there are plugins to limit this for the individual user or in general. Only editors and administrators may use the other categories. There are also plugins who remove certain categories from being shown at the home/front page. So they may only be show in an archive or category overview.
Then make a custom menu item, or a link anywhere, pointing to that category, using a descriptive link text. This category will show all the contributions and only these contributions.
Let new users self register, and set the proper default role to achieve this. Spam posts may be a risk if author is the default role.
There are also “shout box” plugins and talk/chat plugins that will enable posting some content from the front, in a sidebar or widget.
Test the functionality and then invite visitors to contribute.
