RStudio Server Pro (which RStudio Cloud uses behind the scenes) supports collaborative editing. When using collaborative editing, two user's can share the same RStudio session enabling them to share work in the manner you described (similar to Google Docs). This feature is something we would love to get working on RStudio Cloud, but it presents some technical challenges that we've yet to overcome. Rest assured however, that this is something that we intend to have working at some point.
If you want access to this right now, I suggest you consider checking out RStudio Server Pro, perhaps by hosting it on your own server, or on a cloud provider of your choice (we even have pre-built images that should be easy to get started with if you go that route).
For more info: