For 3 centuries horn instructors have advocated using a mouthpiece placement of two-thirds upper lip and one-third lower lip. Iin the case of a shofar I find this technique valid. The fleshy part of the upper lip is the area which determiness the quality of sound. Therefore this upper lip musculature should be developed. A large proportion of upper lip is also beneficial in playing the whole range of the shofar, which is typically two octaves (typically a kudu shofar has greater range); too little upper lip will not allow for the lowest notes on the horn.
Preferably you should blow the shofar from the right side, if possible, because the Talmud says the Satan sits on the right in a bid to condemn the shofar blower (Psalms 47:6).
There is no halachah on how to go about playing a shofar. By inference, if you find it uncomfortable to play the shofar from the right side of the mouth, then you should play on the left side (see Mishnah Berura, 585:6).
Some people play the shofar like they would play a brass instrument, from the center of their lips. While this is an unusual technique, it is permissible. However, the more conventional way to place the mouthpiece against the lips is to position it at one side of your mouth because it is smaller.
In the case of a Yemenite shofar, you will probably want to experiment with both straight and side mouth positioning.
Thanks to veteran shofar player Arthur Finkle for helping provide the material for this article.