On windows I managed to have two TeXstudio sessions in two different processes by modifying the program shortcut (I keep mine on the taskbar) with what Tim Hoffmann said: you have to right click on the shortcut, go to properties and then on the shortcut tab you have to add
--start-always in the Target field.
The shortcut's Target field should look like this:
"C:\Program Files (x86)\TeXstudio\texstudio.exe" --start-always
Note that there is a space between the
--! That is important!
Now all you have to do is start the program twice and work on two different files. I suspect that working on the same file twice will create problems with the auxiliary files so one should be a bit careful. I work on two different TeX files in the same directory but they have different names so there is no overlap.
I have no idea how to do this in Linux but I suspect something similar should be possible!