In my system, i am having both Public and Desktop version of tableau.
I think you should be able to have both installed, could you check you've not installed in a different location?
Sorry for the delay.
Just finished trying with IT and installing Tableau Public 2018.2 in the same location where Tableau Desktop 10.2 is installed or in a different one, uninstalls the 10.2 version.
Any action to follow?