At home the kings had special privileges. They took the first seats at dinner and were served a double portion. They had front row seats for all the games. They choose the Pythii (envoys to the Oracle at Delphi) and give them money for the Oracle. When a king died, the Spartiates mourned for ten days -- women wailing and beating their brows.
What, then, was the significance of the dual kingship as it related to the Spartan political system and the history of governments? For the former, it was a balancing force which helped perpetuate a 1000 year civilization. For the latter, very little. The Spartan way of life succeeded because the helot system allowed the Spartans to train and be the best warriors of all time. Without that one aspect, the Spartan soldiers would have been part timers like everyone else.