There's the EU-wide ePrivacy directive, explicitly stating (Art. 5):

Member States shall ensure the confidentiality of communications and the related traffic data by means of a public communications network and publicly available electronic communications services, through national legislation. In particular, they shall prohibit listening, tapping, storage or other kinds of interception or surveillance of communications and the related traffic data by persons other than users, without the consent of the users concerned

There's also other EU-wide stuff, like:

Charter of Fundamental Rights of the European Union (Art. 7):

Everyone has the right to respect for his or her private and family life, home and communications.

European Convention on Human Rights (Art. 6):

Everyone has the right to respect for his private and family life, his home and his correspondence.

Then there's the UN-wide Universal Declaration on Human Rights (Art. 12):

No one shall be subjected to arbitrary interference with his privacy, family, home or correspondence

And as if that wasn't enough, most member states have confidentiality of communications codified in their constitutions. They are:

Belgium (Art. 29):

The confidentiality of letters is inviolable.

Bulgaria (Art. 34):

The freedom and confidentiality of correspondence and all other communications shall be inviolable.

Croatia (Art. 36)*:

Freedom and privacy of correspondence and all other forms of communication shall be guaranteed and inviolable.

Cyprus (Art 17)*:

Every person has the right to respect for, and to the secrecy of, his correspondence and other communication

Czechia (Art. 13)*:

No one may violate the confidentiality of letters or other papers or records.

Denmark (Art. 72)*:

Any breach of the secrecy that shall be observed in postal, telegraph, and telephone matters, shall not take place

Estonia (Art. 43)*:

Everyone has the right to confidentiality of messages sent or received by him or her by post, telegraph, telephone or other commonly used means

Finland (Sec. 10)*:

The secrecy of correspondence, telephony and other confidential communications is inviolable.

Germany (Art. 10)*:

Secrecy of the mail as well as secrecy of the post and telecommunications shall be inviolable"

Greece (Art. 19)*:

Secrecy of letters and all other forms of free correspondence or communication shall be absolutely inviolable

Hungary (Art. VI):

Everyone shall have the right to have his or her private and family life, home, communications and good reputation respected."

Italy (Art. 15)*:

Freedom and confidentiality of correspondence and of every other form of communication is inviolable.

Latvia (Art. 96):

Everyone has the right to inviolability of his or her private life, home and correspondence."

Lithuania (Art. 22):

Personal correspondence, telephone conversations, telegraph messages, and other communications shall be inviolable."

Malta (Art. 41):

No person shall be hindered in the enjoyment of [...] freedom from interference with his correspondence.

Poland (Art. 49):

The freedom and privacy of communication shall be ensured.

Portugal (Art. 34):

Personal homes and the secrecy of correspondence and other means of private communication shall be inviolable.

The Netherlands (Art. 13)*:

The privacy of correspondence shall not be violated, [...] The privacy of the telephone and telegraph shall not be violated

Austria, Luxembourg and France are outliers in that I didn't find anything in their constitutions during my brief little search.

European Convention on Human Rights Article 8: Right to Respect for Private Life

Everyone has the right to respect for their private and family life, home and correspondence.

UN Convention on the Rights of the Child, Article 16 (English)

"No child shall be subjected to arbitrary or unlawful interference with his or her privacy, family, home or correspondence, nor to unlawful attacks on his or her honour and reputation."

But why the downvotes? That's an important point the article didn't answer either.

Several EU countries have constitutional prohibitions of mass surveillance of private communication.