In most bass player-piano player situations the pianos left hand is relegated to chordal comping while the right hand is freed up more so for filler type licks or soloing. This is very apparent in jazz trios or quartets. The pianos left hand is hardly ever used for bass lines. When it is used for bass it is usually during a breakdown where it is just solo piano. This is also the case when one poor keyboard player has to play piano while playing horns or strings or possibly organ with the free hand.
I played in a group for a while with a keyboard player that expected the bass player to play exactly what he was doing with his left hand. It was a disaster until we convinced the pianist to stick his left hand in his pocket for a couple of songs. He finally understood!