You can't access the files of others using FTP. You can with SFTP, but to copy the file from their directory to yours still involves downloading the file and uploading it again.
However you can do what you want using CGI or through the shell. Basically the user uploads the files to their directory and then it gets copied to yours through system calls (open/read/write) or commands (cp, mv)
With SFTP/shell/CGI you still have to worry about ownership and permission issues. The owner of a file gets to determine who has permission to read the file. The owner of the file is generally the FTP user that created the file.
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