To carry out an action on a file you need permissions to carry out the action in both the folder, and the state the file resides in. "Local" means the file has not been added to the vault and does not have a state. It is only stored locally on your machine and not the server.
Check out this SOLIDWORKS help file that gives a brief overview on permissions.
Also, once you delete a file it is moved to the folders Recycle bin, to completely remove it from the server you need to "destroy" the file, which is a separate permission. You can destroy the file by either "shift+delete" with the file selected, or opening the recycle bin, right clicking the file and clicking destroy.
Successfully setting up and deploying PDM standard is no simple task, and will take a lot of time and questions. I would highly recommend contacting your VAR and arranging to take the "Administering PDM Standard" course, as it will provide some vital information to you on both setting up a PDM system, and administering it.