When did private media outlets cease to exist? How strong was governmental influence over the headlines, did they dictate them or was there some editorial discretion? How did censorship function?
In short, quite similar to how it operates elsewhere - interesting topics were collected, assigned to stuff writers, raw articles edited, corrected, redacted and then printed.
When did private media outlets cease to exist?
Private media wilted out by late 20s, along with NEP. In early 20s press was really wild - whole specter from ultra-left (e.g. deemed to be too extreme by government) to very anti-soviet publications was present. Gradually under pressure from both society and government consolidation happened.
How strong was governmental influence over the headlines,
From strong (main newspapers like "Pravda", "Izvestya", "Trud") to non-existent (if you discount self-censoring). At war time, a special office of military censorship was established and they did extra censoring of all media (plus personal letters to/from frontlines), culling all military-specific information about serving units, supply routes etc.
did they dictate them or was there some editorial discretion?
Main articles for Pravda were heavily editorialized by CPSU leadership. But dictating headlines all the time would be too much of micromanagement. CPSU just made sure that newspaper editors were properly selected and vetted. And those editors were ones who get their careers on the line when anything inappropriate was discovered, so they were well-motivated to do the bulk of internal censoring.
How did censorship function?
Censorship was performed by ГлавЛит - Главное управление по делам литературы и издательств = GlavLit, Main Office of literature and publishing affairs - note that English Wiki translation of GlavLit is rather frivolous. Originally only state secrets were censored out and GlavLit was able to do all work in it's office. As media grew, GlavLit logistically wasn't able to handle all censoring in single place without long delays. And representatives were assigned for every major publication. For small (e.g. local) publications censoring was done by local party operatives in sometimes sloppy and casual manner - or with much zeal, depended really on personalities.
Texts were submitted for review (internal boards inside large newspapers, to local authorities at "area" or "city" level) and then redacted as needed or removed from printing. For serious issues, offending media outlet could be recommended for closure/dismissal of some personnel and it could be first step to political prosecution.
Almost always some redaction happened. In general, it was just an extension of regular editorial workflow, with one extra step - get approval for content.
Note that censoring wasn't limited to ideological topics (like "propaganda of capitalism", "cowering before West", "blackening of soviet achievements", removal of "people's enemy" mentions etc.) - "indecency", "offensive language", "inter-ethnic issues", "sensationalism", "local sensibility", "unproven hearsay" were also checked for and weeded out.
In late 70s, all large texts must be vetted by special representative of GlavLit. Process was called litovka in media slang. Songs for public performance, TV broadcasts must be also vetted out.
Sources
Online book - Martin Dewhirst. Censorship in Russia, 1991 and 2001 Comparison of media censorship at several points - early USSR, 1991 and 2001.
russian Wiki - A timeline of state censoring developments in USSR Gives decent high-level outlook on developments in censoring
Arlen Victorovich Blum - Orwell's Travels to the country of bolsheviks. A. V. Blum was a researcher, who studied censorship in USSR. This is an example of how censoring Orwell worked. He wrote extensively about censorship, see a list of his works on russian wiki page.
russian - Goryaeva. Political censorship in USSR, 1917-1991 Another censorship reseacher, detailed monograph in russian. Review is here.