Somebody can chime in here if I'm off base. I think with Open Source RStudio using the client in the browser is analogous to running RStudio desktop from the host machine. Meaning that only one user can see their own instance. However multiple users could be logged in to different instances at the same time. But each user gets an R & RStudio session running with their login credentials and permissions. (in a standard multi user way).
RStudio pro, however, adds "collaborative editing" which allows two users to share a script and have a Google Documents like experience where they can simultaneously edit a document and see each other's cursor and see changes made by each other. I have not used the Pro version but ran across this feature in some of the documentation.