Perhaps you could maintain whatever database of information you require your users to contribute to on the bot itself, with Tcl binds/commands to facilitate additions to and deletions from the database.
You could then use other binds/commands, perhaps available to the bot owner only, to automatically transfer information to another server by invoking the shell's command line ftp client (using Tcl exec command) to upload. The host/user/pass for the other server would be written into the script, hence only visible to persons with access to your shell (and therefore the scripts).
But, to answer your question, I don't know another way of transferring files between servers without using ftp.
From reading your post, I make the assumption that you've got root access to your server (or equivalent).
In such case, it would be a "mere" matter of installing server software for whatever transfer protocol you choose to use.
Another approach could be to allow anonymous ftp, or grant each user their own ftp account.
Other approaches could be a simple cgi-script to handle a file upload from your favourite browser. This could fairly easily be supported within your eggdrop through some creative scripting.
Yet another way could be using public key authentication with ssh/scp/sftp, although you probably would like to grant each user their own account here aswell.