No, definitely not normal.  Since you're using Arch Linux, I'll assume you're familiar enough with the FS to understand what I'm about to say:

Armory installation directory:  If you checked out from git, it's the git directory.  If it was installed with a package, probably /usr/share/armory.  This is the directory that holds the source code, and all the .py files that actually execute the app/GUI.  Clearly you need read access to this directory, but it sounds like you do, since it runs without root access.
~/.bitcoin directory:  This is the directory that holds all the data from Bitcoin-Qt, including the blkXXXX.dat files needed by Armory.  But Armory only needs to read the files in this directory, it never writes anything.  Say, if you run Bitcoin-Qt with sudo, then it's possible it is setting permissions on the blkXXXX.dat files that allow only root to read them.  That would be a very good explanation for what you see, though not very likely.  Do you run Bitcoin-Qt with sudo?
~/.armory directory:  This directory needs both read and write access.  This is where all the settings and wallet files are kept.  This directory is also created the first time Armory runs. If you ran Armory the first time with sudo, it is possible this directory was created with root permissions.  This would prevent Armory from doing quite a few things, though I expect Armory would flat out crash if it couldn't write the .armory dir... but weirder things have happened.

Also, please run the latest version:  version 0.84.5 (it's the current head of the "threading" branch).  This version has a lot more auto-detection capability and tells you what is missing (if it can figure it out).  If it does work, it will help narrow this down...