Can't speak for Linux, but I install apps on OS X as a non-privileged user all the time. Very few OS X apps actually even have installs, typically you drag the application to where you want to install it and you're done.
Each user has their own ~/Applications folder which, by default, they can do whatever they like in. I think, by default the root Applications folder is only writable with admin privileges, but it in no way precludes you from 'installing' and running stuff in any directory you do have write permissions for.
Can't speak for Linux, but I install apps on OS X as a non-privileged user all the time. Very few OS X apps actually even have installs, typically you drag the application to where you want to install it and you're done.
Each user has their own ~/Applications folder which, by default, they can do whatever they like in. I think, by default the root Applications folder is only writable with admin privileges, but it in no way precludes you from 'installing' and running stuff in any directory you do have write permissions for.