The operating agency would typically be responsible, same as with the red yellow green light liability. There are established protocols and systems but voluntary adoption is incredibly slow. The latest push only got about 10% of the very modest goal of 2,000 traffic signals broadcasting by 2020. https://transportationops.org/spatchallenge/resources/Implementation-Guide

There are many challenges both technical, funding, and end user adoption. There is no mandate for this either at the traffic signal or for equipment in new vehicles to receive it. For the cities, they need to geographically map their lanes and intersections to the lights. Like here's the 4 lanes and the left arrow is for the left 2 lanes. And keep it up to date with any changes. That isn't hard, but it takes money to hire a survey team and someone to program it each time.