In the UK the answer is: not always.
Article 8(1) of the European Convention on Human Rights (ECHR) states "everyone has a right of respect for his private and family life, his home and his correspondence". However, this is not an absolute right: it is a qualified right and interference with such privacy may be justified. Article 8(2) provides that a public authority shall not interfere with the exercise of the right to privacy "except such as is in accordance with the law and is necessary in a democratic society:
• in the interests of national security, public safety or the economic well-being of the country;
• for the prevention of disorder or crime;
• for the protection of health or morals;
• for the protection of the rights and freedoms of others."
The Human Rights Act 1998 (HRA 1988) incorporates Article 8 of the ECHR into UK law and all “public authorities” are expressly subject to the HRA 1998.
The definition of "public authorities" is very broad, going well beyond the narrow categories of central and local government and encompassing any person who carries out functions of a "public nature".
Therefore, if the scenario applicable to Mrs Clinton in the US were to be transferred to the UK, the argument adopted by Mrs Clinton may have little chances of success in the light of the said provisions.
Hillary Clinton failed to quell mounting criticism over her controversial private email account on Tuesday evening after her office suggested she had erased more than half of her emails before turning them over for release to the American public. In a statement released after a press conference intended to end a week-long controversy, Clinton’s office said that she did not preserve 31,830 of the 62,320 emails she sent and received while serving as Barack Obama’s secretary of state from 2009 to 2013. After her work-related emails were identified and preserved, Secretary Clinton chose not to keep her private, personal emails that were not federal records,” her office said, in a defiant nine-page explanation for the unusual arrangement that has put her under political fire.