You can find out wchich editors can edit a particular category, but it will take a bit of effort -- there is no list for each category.
Look at the bottom of the category page. For the sake of discussion, take
http://dmoz.org/Games/Video_Games/Recreation/Browser_Based/Board_Games/Mornington_Crescent/ . At the bottom of that page it says "Category editor: nea". So you know that nea can edit in that category. Move "up" to
http://dmoz.org/Games/Video_Games/Recreation/Browser_Based/Board_Games/ . At the bottom it says "Volunteer to edit this category." No new name to add to the list, then. Move "up" again, to
http://dmoz.org/Games/Video_Games/Recreation/Browser_Based/ . "Category editor: julies" Add julies to the list of editors who can edit Games/.../Mornington_Crescent/. One level up -- no named editors (and at this level it is not possible to become a new editor, so there is no "volunteer" text). One level up -- named editors for Games/Video_Games/ are bluesplinter, deadsea, elminster, lufiaguy, wrathchild, reginaman, lavendergreen. At the top level, Games, add enarra, thehelper, jtaylorj, sidjf. So far, the list of editors who can edit Games/.../Mornington_Crescent/ is enarra, thehelper, jtaylorj, sidjf, bluesplinter, deadsea, elminster, lufiaguy, wrathchild, reginaman, lavendergreen, julies and nea.
In addition to these people, there's a whole large list of editors with editall and meta permissions (well, some of those already named are editalls and metas who are also named in a category). And when I say "large list", I mean 246 people as of right now.
All these people have the same editing rights in every category in the directory. (Except for the Kids&Teens part, which is separate with separate editing rights.) New editors are sometimes surprised or annoyed when other editors come in and make edits in "their" category; it's not all that obvious that just because an editor is named at the bottom of the category he or she is not the sole person responsible for the state of the category, and does not have a greater right to edit there than anybody else. And as a non-editor it can be difficult to realise that the editor named in a category is not necessarily the one who made the latest edits there, and might very well not be the next person to edit there, either.
So the answer to your question is: Yes, it is possible to find out exactly who is able to edit in a particular category. However, this knowledge is not particularly useful, since it doesn't tell you (or us) anything about who out of 250+ people is actually editing there. (And the data about who has made which edit in the past is not public.)