As far as I understand, only your employer can see if you pay the Church tax in Germany, as the employer detracts the tax from your salary.
Can the Church actually see if I pay the tax, and how?
Who else can access this information?
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Whether someone is a member of a church that may raise taxes (öffentlich-rechtliche Religionsgemeinschaft) is stored in the register of residents (Melderegister) with the municipality they are living in. The churches are notified of all changes to said register that pertain to their members; each church will keep a register of members based on these notifications from the municipalities.
Employers have access to this and some other pieces of information that are relevant for tax purposes via a system called ELStAM. Accordingly, they are required by law to withhold a certain part of the salary and pay it to the tax office, which will forward the money to the employee’s church. This ensures that the employee can’t fail to fulfill their tax duties.
Therefore, although there is no direct communication between the employer and the employee’s church, the church can be sure that all their members pay taxes (if they are employed). It will not normally know how much each of its members paid in taxes, nor, equivalently, what their salary was. The tax office, of course, does know this.