Does drush need to be installed as root?
NOTE The issue described in the Question has now be fixed in the latest Drush version.

No, Drush does not need to be installed as root.

I've submitted Drush pull #2246 to update the current Drush install instructions. 

This install method fixes the issue as per Drush pull #2246:

# Download latest stable release using the code below or browse to github.com/drush-ops/drush/releases.
php -r "readfile('http://files.drush.org/drush.phar');" &gt; /tmp/drush
# Or use our upcoming release: php -r "readfile('http://files.drush.org/drush-unstable.phar');" &gt; /tmp/drush

# Test your install.
php /tmp/drush core-status

# Make `drush` executable as a command from anywhere. Destination can be anywhere on $PATH.
chmod +x /tmp/drush
sudo mv /tmp/drush /usr/local/bin

# Optional. Enrich the bash startup file with completion and aliases.
drush init