How to create a basic button to write new entry

Specific pages can be password protected. In the post/page editor, click on the status item in the settings sidebar. There’s a checkmark there you can use to password protect a page. As this is a shared password for anyone allowed to access the page, IMO it’s not a very good scheme. Shared passwords are never a good idea.

You’re better off requiring registration and assigning to those users who can access the page a special capability or role. Then users can manage their own password without impacting other users. However, for this special capability to be effective, you’ll need either a custom template or pattern which enforces limited access. Alternately, make the page a custom post type which has special capabilities built into its schema.

You likely need a custom template or pattern anyway to allow users to add their own listings to the page. Or maybe you could leverage the built-in comments scheme to let users add their listings as comments.