Make sure you are copying the file when you drag it to the other user's Drop Box. You can tell by the small "+" icon that appears next to your cursor.
In all versions of Mac OS X, if the Drop Box is on a different volume (another hard drive or server volume) you will automatically copy when you drag a file to the Drop Box.
With Mac OS X 10.5.2 or later, you will automatically copy when you drag a file to the Drop Box, regardless of where the Drop Box is.
In Mac OS X 10.5.1 or earlier, by default you will move the file when you drag it to a Drop Box which is on the same volume as the original file. This usually occurs when the other user's home directory is on the same hard drive or server as your own. In this case, press the Option key as you drag the item into the other user's Drop Box.
If you have an item in your Drop Box, but you can't open it because it is still owned by the person who moved it there, Get Info on the file and make yourself the Owner of the file. Read more about changing permissions.