The simple answer is Yes.First some work in Joomla.
I would advise you create a new User Group - I usually call UploaderUG - with Registered as its parent. Also create a new Access level, say UploaderView, and assigng the UploaderUG group, and if you wish SuperUser group, to it. In the Create Download Menu item set its acces to UploaderView so that only relevant users see the menu item.
Now in jD UserGroups select the UploaderUG and set its ranking to say 125 - setting the ranking high is critical. Now if you look at the Group Creation/Editing Settings you will find a whole set of things that you can select for members of the UploaderUG.
The final step is to handle the Permissions. Here go to jD Categories and just select the top level Cats. Using the Search tools and selecting jus 1 level is the simplest way . For each top Level Cat click on the Permissiuons tab. Select the UploaderUG each time - it is easy to inadvertently select the Public UG. Set Permissions Create, Edit, Edit State, Edit Own to Allowed (I rarely advise setting Delete). As you select each one a green tick appears. Click on SAVE not Save & Close. You will then see what has been set. Now if all is ok you can do Save & Close. Note you should see that Download Permission is already Allowed(Inherited).The Permissions will automatically cascade down to the Downloads. (NB NEvER set deny permission!!!).
There are several articles on User Groups & Permissions https://www.jdownloads.net/index.php?option=com_content&view=category&id=32
If you need a more advanced arrangement with different users allowed to edit/create then that can be resolved later.