Legendary Super Saiyan requires one to have consciousness while maintaining the form. What we have seen so far (canon and non-canon) was the berserk state, where the user is simply out of control.
As we know in order to attain Super Saiyan 4 one needs to have consciousness during the Golden Great Ape state. The Great Ape forms have been known for all Saiyans to be a berserk state that can only be triggered when specific conditions have been met. The user is "rewarded" with tremendous power, but no control. This all is extremely similar to Legendary Super Saiyan Berserk. Once the user gets consciousness in the state he/she can then evolve into the next stage, which would be Super Saiyan 4 (for Golden Great Ape) or Legendary Super Saiyan (for Legendary Super Saiyan Berserk).
Since both forms (LSSJ and SSJ4) require the same amount of consciousness and mental fortitude, does this mean that they are basically the same? I found it extremely difficult to find some actual information on the progress of the two being unlocked so therefor this conclusion is purely theoretical.
Assuming they are, all Saiyans are able to become a Super Saiyan 4 with enough training, consciousness etc. Are all Saiyans also able to become a Legendary Super Saiyan or is it purely genetic? From what I've seen so far, the only distinctive features it has, is emitting enormous amounts of ki. We have seen that Saiyans were leaking ki all over the place, so in that sense it should be possible for all Saiyans eventually, or am I completely wrong?
Sources: mainly dragonball wikia, Reddit, the series from which DBGT for knowledge about SSJ4, DBZ+DBS for knowledge about LSSJ and DBS for Whis explaining loads of things.