Separation of Church and State is one of those quips that seems like eternal wisdom and something that everybody knows to be established in law. Additionally to that, we all "know" that it is primarily for the protection of Church against State regulation of religion. So, who first declared "Separation of Church and State" - Church or State?
In the natural law, the hierarchy of legitimate authority on all matters of morality and human interaction, which is the total set of areas of Church and State action, flows downward, starting with the One True God. Definitions of right and wrong begin and end here.
Jesus Christ declared to his apostles, following His resurrection, "All power is given to me in heaven and in earth. Going therefore, teach ye all nations; baptizing them in the name of the Father, and of the Son, and of the Holy Ghost. Teaching them to observe all things whatsoever I have commanded you: and behold I am with you all days, even to the consummation of the world." Matt 28: 18-20 (Emphasis mine)
That is the great commission. The apostles are the Church and they are commanded to teach the gospel of Christ to "all nations" and they have the power and authority of the Lord of All with them "all days", who has "all power". Does this sound like someone who needs protecting?
The Church is the ante-chamber to heaven. She teaches right from wrong and holds the "keys to the kingdom of heaven", and holds the authority to bind or forgive a man's sins. Her one concern and raison d'etre is the well being of the People and their eternal salvation. Matt: 16: 19 & John 20: 23
On the next lower tier of earthly authority we find the People. Each and every person born and alive is subject to God, Christ and the Church - everybody, and it doesn't matter weather they know it or not, or believe it or not - everybody is subject to this authority, for this chain of authority is the only possible way to heaven, as set by God, Himself. It is here, at the People level, that the State is created, for it is a construct of the People, for the People. It is impossible for anyone to confer authority to another which one does not himself possess. So, if the People, who create the State, are subject to the Church in all matters of faith and mores, the State is bound to the same teachings and authority. In a harmonious world, the State would form it's laws and civil structures to compliment and support the teachings of the Church, much as a Mother supports and backs the Father as he teaches their children the way to live; for in unity, we find harmony.
Unfortunately, the State, being a human construct, and not having the guarantee of the Lord being with them "all days" is subject to horrible and destructive errors. For that reason, it is paramount that the People recognize these dangers and always form their State in specific subjugation to the Church. The United States Constitution was written that way, albeit for the wrong reasons and in a clumsy fashion. Remembering the religious persecutions of the English and other European governments, while said governments claimed to hold the authority of the Church, the founders specifically proscribed government interference to religion with the First Amendment. The grave error lies in the idea that the People have the right and authority to create forms of religion, which suit their tastes, in defiance to the legitimate authority of the One True Holy Catholic and Apostolic Church. Thus it is through the rebellion of the People that the rebellion of the State is fermented.
To answer the question above, I would say that it is the State which declares the separation, for they are the ones seeking independence, and in fact, there is no difference between the English king proclaiming himself to be the "head of the church" and the founders of the Unites States declaring that Church is separate and holds no authority over the State.
Separation of Church and State is like separation of wife and husband, or separation of players and umpires, or separation of enlisted men and generals; it is a declaration of insubordination and only leads to chaos.
One excellent example of this insubordination is the idea of marital divorce. The Catholic Church has declared that divorce is impossible. A couple go to the Church and apply for marriage and upon legitimate approval, marry. Later, they fight and wish to divorce. They know better than to go back to the Church for a divorce, so they go down the tier of authority to the State, and the State, not recognizing their subjugation, grant a "divorce". In truth, they are not divorced at all, and the only legitimate thing the State can do for them is to make ruling on how to separate property as they remain physically separated, yet truly married - for life.
We are learning, the hard way, that there is never a true separation of Church and State, and that the only two possible configurations in this relationship are ordered or disordered; subjugated State or insubordinate State. Examples of errors include laws and government actions on divorces, contraception, abortion, homo-sexuality, extra-marital co-habitation, rest-rooms, property seizures, currency debasement, power hoarding, etc. These are all symptoms of our disordered and separated State and they all trample upon the legitimate authority of the Church and the intrinsic right of man to find his way to his God.