In right to privacy, to is an ordinary preposition, not an infinitive marker.
This can be confusing for learners, because right takes four sorts of complement:
- infinitive clauses marked with to ... the right to speak freely
- preposition phrases headed by of ... the right of free speech
- preposition phrases headed by to ... the right to free speech
- preposition phrases headed by in ... my rights in this property
All of these have essentially the same meaning; the choice between them will usually be driven by how the writer wants to characterize the right within her larger discourse.
The infinitival complement is employed to speak of an active right, a right to perform some action. ... You have the right to remain silent: you cannot be compelled to speak.
The preposition phrase with of is usually employed to speak of an abstract right, a right considered as an enduring property or institution. ... The right of peaceful assembly is fundamental to democracy.
(Note, however, that the 'infinitival' sense can be married to the of sense by using a gerund as object: the right of assembling peacefully.)
The preposition phrase with to is usually employed when speaking of an actively asserted right, a right put forward in the face of opposition. ... This statute violates plaintiff's right to privacy.
The preposition phrase with in is usually employed in civil law to speak of a financial or directive interest with respect to a specific property. ... Her right in the family firm extends only to participating in the profits, not to management.
Of course other preposition phrases can be used with right—rights under the Constitution, rights for women and children, rights beyond those enumerated—but these are adjunctive, modifiers, rather than complements.
In your specific example, the "right to privacy" is not the right to "have" private matters in your emails—it is the right to insist that those matters remain private, not subject to inspection by your employer.