Deleting another user's directories from my own

In order to delete a file in a directory, you need to have write permission to that directory. Obviously the users are creating directories without you having write permission, in which case the answer to your question is no, you can't do anything without talking to the user in question or the sysadmin.

However, if I was the sysadmin, I would be worried about what you're doing. It is not normal for users to be creating files in other users directories. Is there a better way of doing what you're doing? For example, uploading to a suitable web application would give better control, and also mean that you don't have to worry about this.