A daemon process running as a special user has created files which only its user can read. The directory is owned by me and I have write access to the directory. Is there any way that I can read or copy the files without sudo rights?
You do not have the permission to read it - therefore, you cannot read it.
You have permissions to change the directory - you can rename or remove the file.
In a way, you can access the file's name, but not its content.
If it's a huge deal, you could just boot a live disc to read it in a round-about way...
Boot into the Live Disc (root login by default) and (all in the terminal):
That'll get you the contents - if it matters enough to do the process.