Do I have to use a backing sheet?
Yes! The backing sheet forces the blade to produce a better cut by supporting the face paper of the mat you are cutting. The cutting is always done from the back of the mat board which means the last thing the blade has to slice through is the colored surface paper of the mat board. Logan board mounted mat cutters all have thin slots cut down the length of the base board to be used when straight cutting. But when bevel cutting this slot may allow the face paper to be "pushed down" by the blade tip causing the blade to pop through and tear its way across. With a scrap piece of mat board underneath, the blade cannot push the face paper down at all. Instead it slices cleanly through resulting in a better cut.
Can I use something other than mat board for a backing sheet?
Matte board is the most appropriate thing to use as a backing sheet. Anything else will prematurely dull your blade. Mat board is firm enough to support face paper of the mat being cut along with being soft enough not to dull the blade or cause the blade to flex causing hooks or curves. It will however have to be changed periodically. Letting a backing sheet get too chewed up from use can also cause hooks or curves in the cut. The question has been asked if self-healing vinyl desk covers could be used. It sounds good, it lasts forever and never has to be changed but the answer is "no". That material is too tough for a blade even to score across. It would require much more force to be cut on, cause blade flexing and most definitely dull the blade faster than usual.
Lastly, it is important that the backing sheet is at least as long as the mat you are cutting and at least 3 to 4 inches wide. If there are any areas under the mat where the blade will be passing that is not supported by backing sheet, the blade will probably not cut through.
Follow this advice and better mats are guaranteed. HAPPY MATTING!