First of all, please no. Just don't.
And no, it will not be effective. Even given that there are some people that can tow trains by ropes in their teeth, you cannot clench your jaw enough to be able to hold on to it if someone hits the blade. It will spin out, and if you are lucky, leave your teeth behind.
Second, despite what they show in cartoons, you are much more likely to hit the sword yourself, either on the way through for a strike or when drawing back. Add to that, you can't really generate enough force to cut deep enough to do the damage you want merely by twisting your neck around, unless you get lucky.
So in effect, you are risking a lot of damage and limiting your effective attack. Even your right hand sword (as shown in the picture) will be of limited effectiveness since you are only using one hand and missing the "push pull" action that gives your strikes the most force.