IMHO the desktop is just 1 location (folder) for each user, plus 1 location system-wide ('All Users' desktop folder) that are merged into the view that you have on your screens. Just the coordinates of the short-cut-icon determines where it is 'located'.
AFAIK, new icons can't be easily 'steered' into a specific position. Windows just picks the first available position (top to bottom, left to right on LTR (left to right writing) Windows) on the main desktop.
- Create folders on your desktop for each monitor you have (DL-Screen-left, DL-Screen-middle, DL-Screen-right)
- Drag each folder to the corresponding monitor
- Download any file into the correct folder, your initial download location should be 'Desktop', so you can browse easily into the right sub-folder
To make the last item easier, you could use separate browser-brands on each monitor (IE, FF, Chrome, Opera), but you'd have a bit of trouble keeping favorites synched.