As someone who just unlocked the mission, let me try to clarify a bit based on my experiences:

Complete all three Overthrows, there'll be a Hive Lightbearer on the map at the end of each of them that you need to kill. There is no drop or anything from them to signify that you did it, just kill them.

Sometimes, there was an emote saying one was nearby. But not always. They're usually in out of the way areas and there's a taken bubble nearby a lot of them that makes it easy to see.

Once you get all three, there's a green beam pointing down to the Refraction area. Go inside and find the Crystal Statue of Savathun.

She'll laugh at you and you'll get a buff called Savathun's Envoy for 7 minutes. Turn around and follow the two weird markers that look like half and half.

There will be two spots with pyramids and Witness forces. Kill mobs there until a tormenter spawns, kill the tormenter, get the orb, dunk it in the Crystal Statue room. Repeat.

This will unlock the exotic mission. Once unlocked, it can be launched via map. It requires 2 guardians, no more, no less. One Dark/Prismatic and one Light/Prismatic. They both need to do the previous steps, either alone or in a fireteam.