Toothpaste might be similar to the final compounding step that polishes the headlights after going thru the sanding/cleaning process. Can't see it being effective without using a polishing disk with a cordless drill. Then again, toothpaste might be too aggressive for this step as it's a paste rather the typical liquid polish found in most headlight cleaning kits.

Per other feedback, I'd go ahead & buy the headlight cleaning kit. Cost you somewhere around $20-$35 depending on whose you buy. The 3M kit I used took care of my wife's VW's headlights in about a 3 hour effort.