This looks strange.
We use also RStudio Server Pro, and when you open multiple sessions, they are each one with independant R process.
Now, they all share the same ressources on the machine, so there could be some slow down if the computation is intensive (ex. one process launch a code on all the core), but if you environment are correctly sized, this should not be the case. 
How are you opening the new session ?
You can go to <yourserveradresse>/home to get a peek at all you opened sessions and there status (idle, suspended, running, ...)