BuddyPress.org

“how can multiple users write in that blog under their own names?”
Right. Basic WordPress – https://codex.wordpress.org/Roles_and_Capabilities#Contributor

“but how can they enter the writing zone? “
Simplest way – if your users are logged in your BP install as contributors, they can go to bp-admin-bar at top and move cursor to “My Blogs”, which has a dropdown to list of blogs. When they hover over JuConnect link, the flyout nav will show the link “Add New Post” where they will be brought to the Post > New Write panel.