I'm using Linux Mint, and Unity has never asked for root permissions. I only needed root to install the deb package via dkpg. Which Linux distribution are you using?
used the Unofficial Platform-Agnostic Installer and the "installer" wants root i give him root => next he extracts the files (in current directory by default) after the "installation" i can launch unity without root. but what does the installer do to requiere root?
sry, found it in the comments: http://blogs.unity3d.com/2015/08/26/unity-comes-to-linux-experimental-build-now-available/ still scarry
I would guess it's because the chrome-sandbox has to be owned by root (but why is this I have no idea... but unity does complain if you change it to your user).