Privacy was the ability and right to protect some aspects of one's life from public scrutiny.
A common phrase was, "to speak with someone in private". (ENT: "Fortunate Son")
According to Captain Jonathan Archer, dating wasn't discouraged aboard Enterprise NX-01 but there wasn't a lot of privacy. He answered this to a question from Geoff Miles in 2151. (ENT: "Breaking the Ice")
During the 23rd century, on Starfleet ships when particularly private conversations needed to be had, the captain had the ability to order the ship's computer to observe their privacy. (DIS: "The Vulcan Hello")
In 2367, the con artist styling herself as "Ardra" attempted to seduce Captain Jean-Luc Picard in his quarters. He attempted both to activate an intruder alert and to contact the USS Enterprise-D's bridge, but neither attempt was successful, and she told him that she had arranged for a bit of privacy. (TNG: "Devil's Due")
That same year, Lieutenant Commander Geordi La Forge disconnected both the audio and visual pickups inside the Enterprise-D's conference room so that Reginald Barclay could not monitor the senior staff after he had created a neural scan interface which transferred much of his higher brain functions into the starboard computer core. (TNG: "The Nth Degree")
The Cairn, a telepathic species, did not understand the concept of privacy, as nothing among them was hidden. In 2370, when Maques detected a part of Lwaxana Troi that was "dark," her daughter Deanna Troi initially believed mistakenly that Maques was referring to privacy, not realizing that he had sensed her mother's metaconscious, in which she had suppressed her memory of her lost daughter Kestra. (TNG: "Dark Page")
In 2372, Lieutenant Commander Worf argued to Lieutenant Commander Jadzia Dax that Constable Odo valued his privacy and did not like to socialize. Dax was of the opinion that this was only an act, but Worf stated that it wasn't, that he told him so himself. Dax then noted that Worf was socializing with Odo when he said it. (DS9: "Broken Link")
During the 24th century, Federation citizens using recreational holoprograms had some expectancy of privacy. (DS9: "Our Man Bashir") In 2373, when Tom Paris discovered B'Elanna Torres using the program Insurrection Alpha, she asked him if he was in the habit of walking in on people's private holodeck time. He reminded her that they had a lunch date and, referencing the unusual contents of the program, stated that he had expected to find her shooting pool at Sandrine's, rather than hanging out with Seska. (VOY: "Worst Case Scenario")