laying on the right side or on the back squishes the vena cave, a major blood vessel that carries the newly oxygenated blood through the body. the uterus is too heavy at this point, so that could be the problem with the asphyxiation feeling. laying on the left side promotes the best blood flow.
as far as laying on the left side and sometimes feeling unable to breathe, i too have had this problem, and have found that it's my arms. if my right arm is hanging over my chest, as opposed to along my side, or if my left arm is in a weird position, then i can't breathe right. i feel like my chest is sort of folding in on itself when my arms are positioned incorrectly.
something that kinda helps is a pillow in front under my belly, one b/w my knees, and another one along my back, so if i roll onto my back i don't suffocate as bad since there is something under my right side.