Is it doing that in the same RStudio cloud RStudio session ? because I believed a package installed in one session would be still available in this session if you close and reopens it. However, it you start a new session, it will go fresh everytime. Each sessions are independant. So if another user use the same script as yours with checkpoint, but in a new session, it will reinstall all from the checkpoint url, but reinstall it no matter what.
This is because I believe checkpoint("2020-07-16") will just point to a snapshot of the cran server at this date, and you'll be able to install package from this date. But it won't cache package for others session.
Know that you can achieve the same with https://packagemanager.rstudio.com by using one of the snapshot url. See https://packagemanager.rstudio.com/client/#/repos/2/overview
You can for example use this url to use a snapshot of the 2020-07-06:
options(repos = c(CRAN= "https://packagemanager.rstudio.com/cran/299"))
One advantage for rstudio cloud is that you can have linux binaries available, which will make all the packages a lot quicker to install ( very fast !!). You would use this url in this case:
options(repos = c(CRAN= "https://packagemanager.rstudio.com/cran/__linux__/bionic/299"))
If everyone is using the same CRAN url for a project, the package version will stay the same. Like with the checkpoint package. But, the binary availability feature is really a game changer for unix environment ! so quick to install a bunch of package. So it will be a lot easier for your other users to install the same environment. (And I believe Rstudio cloud is using linux binaries with latest cran version by default, so you see easily how fast it is).
If you did not know about Public Package Manager, here the announcement post:
Hope it helps with your use case.