The modern concept came out of the Disputes between the Holy Roman Emperors of the High Middle Ages and the Popes. Which was superior and when? Both claimed ancient rights to be independent of the other, but at the same time they had to work together. From is medieval dispute came the concept that religion and the State should be two different organization (Through the modern Concept of both being independent of each other came out of the Enlightenment of the 1700s and true independence did not occur till AFTER the invention of the modern pulp paper and thus cheap newspapers about 1850 when the State no longer had to use the Church to spread news of Tax increases etc to the people).
In the Eastern Half od Christendom (as that term was used in the Middle Ages) a different rule of Religion and state was used, the Church controlled by the state. This was the rule in the late Roman Empire in the West and later on in the Surviving Eastern Empire (And its successor States of Russia where even under Stalin the Church was part of the state hierarchy and remains so to this day, through in the 1920s some efforts had been made to destroy the Church in Russia, that soon passed and even Stalin used the Church to help him rule).
In the Moslem East, a third set of rules developed, or more precisely the pre-christian rules of Church and state survived, i.e. the Church and State were one and the same. The Caliph was NOT only the political head of the Arab Empire but also its Religious head. His underlings rules not only as the Caliphate Governors but also as the local head of the local Moslem's (i.e. equivalent of being both Governor AND bishop).
Now the Shiite came under a Fourth set of rules. At first they ruled like their Sunni Brothers, but as the Shiites became more and more the minority community within Islam, the more they had to hide the existence of being Shiite. This included following the dominate practice of the area they were living in while maintaining a separate religious AND political group. Unlike the Catholic West where you had two leaders, Pope and Emperor, who fought over where Religion and the state both started and ended, in The Shiite areas, the Shiite were the Political opposition to the Caliph after Ali (Shiite literally means "party of Ali" i.e. they backed Ali as the successor of Mohammad for he was Mohammad nearest male relative, while the Opposition picked the Caliph by traditional arab custom of an election when the previous head of a tribe died. The Sunnis are descendant from these post-Ali Caliphs who the Shiite do NOT recognize as rightful successor to Mohammad. Technically the Sunnis did not firm up as a religious doctrine till the ninth Century (almost 200 years AFTER Mohammad) but the basic concepts was alive and well during the previous 200 years, Church and State were one and the same and on the death of one leader, came the election of his successors by the men of that tribe.
The Shiite opposed this and supported the descendants of Ali till they died out (They is a split in the Shiites between those who say only the first seven descendent's of Ali was rightful rulers of Islam and those who believe the last real decedent of Ali was his 12th descendent's (These groups are often call the "Seveners" and "Twelvers", most Shiites today are Twelvers, but the Assassins of Medieval Times were Seveners). Since the death of the last decedent of Ali, the Shiites are waiting for the return of that last Iman as a leader to take back what is his by right (The leadership of the Moslem World).
Thus the Shiites do NOT believe in Separation of Church and State (as that concept is derived from the fight between the Pope and Emperor NOT between which two groups of Mohammad's follows should rule the Islamic World). On the other hand the Shiites, as the opposition to the Sunnis, has a history of Religious leaders independent from their Political leaders. For example when the Shah ruled Shiite Iran one of the two houses of his Parliament was made up of Shiite religious leaders who would rule on issues of Religion. When the Shah was overthrown this group of Religious leaders assumed the position of power within Iran but they did NOT want to rule on a day to day basis. Thus the present "Islamic Republic" Constitution where the people elect the President and Congress but the final say is in the hands of the Supreme Religious Leader of the Shiites in Iran (Ayatollah Ali Khamenei) NOT the President of Iran (Mahmoud Ahmadinejad), through the President runs most of the day to day business of the Government. The Islamic Republic is a strange mix of Religion and Secular Government NOT a true Theological Government but also not a Secular Government either.
See Global Security view of Iran (GS tends to be strongly Right wing and anti-Iran but if you remember those facts, GS gives a good overview of Iran's Government):
http://www.globalsecurity.org/military/world/iran/leadership.htm