As far as whether you can use the edited resources - many DLCs have a Terms of Use that spells this out specifically, but in the cases where it doesn't, the general expectation is that you are allowed to use edited versions of resources wherever you are allowed to use the resources themselves. This means that you can use them in your own non-commercial games, you can usually use them in your own commercial games (but only as long as the original resources are OK for commercial use), and you generally can not publicly re-post or sell the edited resources for other peoples' use.
As far as whether both people (the game designer and the artist) need to own the DLC in order to have one make edits for the other, the answer is that technically, yes, both people need to purchase the DLC. That honestly seems a little onerous to me, since the two users are using the same resource for one shared purpose (similar to installing it on one computer and having a few different team members use that computer at different times)... but if you want to do it right, you've gotta both buy the DLC (or you need to buy it twice, once for yourself and once for your artist).