You don't need to be carrying the item at all, and indeed are not allowed to be carrying it unless you are actually holding it (e.g. you may not cast it on your hat, or belt buckle, or a dagger in a sheath).
You may cast it on an item you are holding in your hand, or you may cast it on an item that is not being worn or carried. In either of those cases, the spell is on that item and the Darkness persists thereafter.
The DM would have to rule on whether you may cast it on an item you are holding, and then wear or carry it. It would seem strange to me that you cannot cast it on your hat on your head, but could cast it on your hat in your hand and then put the hat on your head and it works. I would, accordingly, rule as a DM that the spell no longer moves with the item if the item is subsequently worn, or carried without being held.
So, to answer your question directly:
You may cast it on an item and leave it somewhere (or throw it, or kick it somewhere) and the Darkness will move with the object and stay where the object stays (as long as concentration is maintained on the spell, of course).
And indeed, I would say that you may cast it on an item and give it to another person, and the Darkness will persist, and travel with the item, as long as the item is held by the other person, and not worn or carried other than held in hand, and concentration is maintained by the original caster. That however, is not explicitly defined, and would be a DM ruling to decide.
If you cast it on an eligible item, and that item becomes no longer eligible (i.e. someone puts it on to wear, or hangs it from his belt, then the DM must decide what happens. He could (a) rule to let the spell persist on the now worn or carried item, (b) rule that the act decouples the item from the spell, and the spell persists, unmoving, at that location, or (c) rule that the act ends the spell on the no-longer-valid target. I would go with (b).