I think what the best way to do this may depend on the number of students involved. It's of course possible to assign a single set to all students and then adjust their due dates on a student-set by student-set basis. I think this is how I would be inclined to approach it in most cases.
The other possibility, which you note, is to just have a copy of the set for each student. This may have the advantage of being able to easily produce a table of sets (in the homework sets editor) so that you can update due dates for all sets at once. So for a lot of students or sets that might be desirable.
The student-set deadline adjustment requires more intensive management of the students' sets; the multiple-set solution requires creating and then managing more separate sets. I think it's not entirely clear which will be best.