The dependence of the copy on the original has some important consequences:
If you change the visual representation of the original, the representation of the copy will change as well. (See Change the Visual Representation)
You cannot delete the original until the copy has also been deleted.
Since the part copy only exists on the client, you cannot save a part copy to disk.
If you want to create a dependent, non-shallow copy of a part, you can perform a merge operation on a single part. This type of copy does now have the same consequences: the resulting copy is basically independent except that it cannot exist without its parent. Or you can simply reload the part to create another copy of it.