No you dont need to install any OS on the laptop, because the alternative OS, including the programs it runs, all run FROM / BY / WITHIN the USB stick.
The USB stick is a "drive" (with memory storage, input / output etc.) just like the hard drive of the laptop. The USB stick does use the processor and RAM of the laptop to do the processing work, the screen for display, the wifi module for internet etc etc.
However, as long as the hard drive of the laptop is not used in any way, it will leave no trace there.
If you are interested in the tekkie stuff, come
HERE and
HERE to do some more reading.
As I said, the major problem is slow speed - no matter how fast your USB stick is, it is not even as fast as the average low end netbooks using Atom processors.