The game is controlled by the augments and Mods on or in your gun in BOTH hands as proven by this simple easy test. Illaster's Magic sun + 1 Fire% damage Fuel, + 1 3% critcal augment, Congrats your Smackdown and Shock and Awe will now both crit, and ignite. It doesn't matter which hand the gun is in or what else augment//mod wise is equipped.
Some augments will not work however with strikes some on hits, I haven't tested them all, but I know the nova's on-hit and death don't proc with strikes, also Shrapnel doesn't either.