How to start a second X session?

I think you can do it with this:

$ startx -- :1

Note that you need to be on a text console. If you do this from an X session, you may not be authorized. First Ctrl+Alt+F1 to switch to a text console and log in there.

Press Ctrl+Alt+F7 and Ctrl+Alt+F8 to switch between the X sessions (the F key numbers may vary depending on your distribution).

If you want more control you can add more options to the command like so:

$ startx gnome-session -- :1 vt8

This will start up gnome-session on display :1 and run it on virtual console 8 (Ctrl+Alt+F8).

I have found that the following way works for me:

sudo xinit /usr/bin/xterm  -e "su $(whoami) -c pekwm"  $* -- :1

What happening here?
First I start xinit with an xterm, as a parameter to that xterm I su to become myself. From there I start the window manager.
Only thing you need to do is minimize (not close) the xterm. Probably you could tell the xterm to start minimized—but I have not care enough to find that out.

Also: Replace pekwm with the session you want to login to. Look in the .dewsktop-file in /usr/share/xsessions

Edited code. Changed from my real username as in example to the output from whoami