How can I run a separate X session on a different physical display?

What you want is a multiseat Xxorg configuration. I don't know which distro you're using, so I'll just link to the xorg Wiki entry. X is well suited for this, since 20+ years ago many institutions did this with all of their Unix machines.

One limitation: You won't be able to use the same keyboard and mouse for both displays.

I just wrote a quick little tip on how to set up and run separate xsessions on 2 monitors. Check it out http://www.nu2upc.com/linux-tips/configuring-dual-displays/

First let me explain what I am going to accomplish to see if this meets your needs. I am going to be running 2 monitors, 1 is connected via VGA and the other by DVI. The VGA monitor is to be connected to screen 0 as monitor 0 on xsession 0, and the DVI being on 1.

Problem that I ran into is that the system was automatically assigning 0 to the DVI monitor therefore not giving me the results that I desired.

Solution was to simply unplug the DVI monitor, reboot and start with a clean xorg.conf file. I opened up nvidia-settings checked to see that my VGA monitor was configured the way I wanted, then I plugged in my DVI monitor, clicked on “X Server Display Configuration” then “Detect Displays” and it found the DVI monitor. I then set DVI monitor to be right of my VGA monitor and clicked on “Configure” and selected “Separate X Screen” then ensured that all other settings were as I like. I clicked on “Save X Configuration File” and said yes to merge the changes. Then I rebooted my system. and viola! I now have my 2 displays running each in a separate xsessions!