In fact. livekd is not realtime kernel debugger. It just make memorydump many times and do on the memorydumps. WinDbg do so too.
It is only softice which can do kernel debugger on one machine.

WinDbg and livekd is a ring3 application. You can never expert it can do ring0 debug on one machine. Because if ring0 paused, no ring3 application can running unless you are in VM.