“Church and state should be separate” entails that the Church should have no power over the state, but not that the state should have no power over the Church.
The Orthosphere
Wherever an altar is found, there civilization exists – Joseph de Maistre
The Orthosphere 
Church is not that kind of entity that seeks power. Thus, certainly Church should have no power over State. It has no power even over Church-goers.
But Church should inform the State. That is, the State must be, and historically all European states have been, molded by the Church. The notion of secularism itself comes from the Church and does not exist in non-Christian lands.
So, the relation of Church and State is like soul and body.
Thus, certainly Church should have no power over State. It has no power even over Church-goers.
What a bunch of modernist silliness.
All legitimate authority (power) is a capacity to create moral obligations in positive law out of the raw material of natural and Divine law. And the Church certainly does so.
If we are talking about illegitimate authority that is a different matter, of course. But there is no reason to obey the State when it attempts to project illegitimate authority other than fear of punishment; and there is an obligation to disobey the State if it attempts to compel an immoral act.
It’s not really a debate. It’s a way to get you to disarm yourself unilaterally, then slouch into oblivion.
Zippy,
Are you equating power with authority?
How is authority recognized?
Maybe a relation of love, perhaps call it friendship, needs to exist between the authority and those subject to the authority.
Or reason alone sufficient?