In the TOS episode "Court Martial" Captain Kirk is tried, including a playback of the visual and audio recordings in the computer log:
(On the courtroom viewscreen)
UHURA: Meteorology reports ion storm upcoming, Captain.
KIRK: We'll need somebody in the pod for readings.
SPOCK: Mister Finney is top of duty roster, Captain.
KIRK: Post him.
SPOCK: Attention, Commander Finney, report to pod for reading on ion plates.
FINNEY [OC]: Message Received.
SPOCK: Officer posted, Captain.
(The ship suddenly judders)
KIRK: Stand by on alert status, Mister Spock.
HANSON: Approaching ion storm, sir.
KIRK: Warp factor one, Mister Hanson.
HANSON: Warp one, sir.
(There's another sharp jerk, and Kirk presses a button on his chair panel.)
SHAW: Reverse. Stop. Go forward with magnification on the panel. Freeze that. Captain Kirk is now signalling a Yellow Alert. Go forward, normal view.
(On courtroom viewscreen.)
UHURA: Call from the pod, sir.
KIRK: Tie in.
FINNEY [OC]: Finney here, Captain. Ion readings in progress.
KIRK: Make it fast, Ben. I may have to go to Red Alert.
FINNEY [OC]: Affirmative.
KIRK: Hold our course, Mister Hanson.
HANSON: Aye, aye, sir. Natural vibrations, force two, Captain. Force three.
KIRK: Engineering, then ion pod.
UHURA: Aye, aye, sir.
CREWMAN [OC]: Engineering.
KIRK: One third more thrust.
CREWMAN [OC]: Working.
FINNEY [OC]: Ion pod.
KIRK: Stand by to get out of there, Ben.
FINNEY [OC]: Aye, aye, sir.
HANSON: Force five, sir.
KIRK: Steady as she goes, Mister Hanson.
(And the close up on the Captain's panel shows...)
SHAW: Freeze that! If the court will notice, the log plainly shows the defendant's finger pressing the jettison button. The condition signal reads Yellow Alert. Not red alert, but simply Yellow Alert. When the pod containing Lieutenant Commander Finney was jettisoned, the emergency did not as yet exist.
KIRK: But that's not the way it happened.
Thus we see that all the time, or under some conditions, the computer logs include sound and images of events happening all over the ship, or at least on the bridge (and other important spaces?). Note that those playing the computer log can have it display images from at least two different angles on the bridge and at different magnification levels.
In Spock's court martial in "The Menagerie, Part 1":
MENDEZ: Why? What does it accomplish to go there or to take Captain Pike there? I want to know why.
SPOCK: Are your comments a part of the record, sir?
MENDEZ: Yes, it's on the record.
SPOCK: Thank you. Request monitor screen be engaged.
MENDEZ: For what purpose?
SPOCK: To comply with the request you just made, sir, that I explain the importance of going to Talos Four.
KIRK: By asking why, you've opened the door to any evidence he may wish to present. Apparently what he had in mind.
MENDEZ: Present your evidence. Screen on.
SPOCK: This is thirteen years ago. The Enterprise and its commander, Captain Christopher Pike.
SPOCK [on screen]: Definitely something out there, Captain, headed this way.
KIRK: Screen off. Chris, was that really you on the screen? (flash) That's impossible. Mister Spock, no vessel makes record tapes in that detail, that perfect. What were we watching?
SPOCK: I cannot tell you at this time, sir.
MENDEZ: Captain Pike, were any record tapes of this nature made during your voyage? (flash, flash) The court is not obliged to view evidence without knowing its source.
SPOCK: Unless the court asks a prisoner why, Commodore. You did ask that question.
Unless "Court Martial" and "The Menagerie Part 1" happen in alternate universes where Starfleet starship record tapes and/or computer logs have great differences in their degree of detail, there must be a difference that Kirk could spot within seconds between computer log visual recordings and the images from Talos IV in "The Menagerie Part 1".
In "Court Martial" the images come from an overhead position and do not switch angles until a reverse angle and magnification are asked for. In "The Menagerie Part 1" the images from Talos IV play like a television episode (like the one they were filmed for and edited into in real life).
I believe the images in "The Menagerie Part 1" are filmed with the cameras at about eye level and cut back and forth between camera angles to show whoever is speaking at the moment. I think that they also start with a tracking shot showing the exterior of the Enterprise zooming in on the bridge and showing the interior of the bridge.
Later on the images from Talos IV show a private discussion between Pike and Dr. Boyce in Pike's cabin. If nobody had questioned the origin of the images until that moment it would indicate that events in starship private quarters were not recorded. but since it was already established that the images shown were not from official recording there is no evidence whether or not official recordings show events in crew cabins.
The Federation Starfleet is not the only organization that makes recordings. In "Errand of Mercy" Kirk and Spock enter the office of Klingon Commander Kor on the planet Organia:
KIRK: Just stay where you are, Commander.
KOR: You have done well to get this far through my guards.
SPOCK: (taking his weapon) I believe you'll find that several of them are no longer in perfect operating condition.
KOR: So, you are here. You will be interested in knowing that a Federation fleet is on its way here at the moment. Our fleet is preparing to meet them.
KIRK: Checkmate, Commander.
KOR: Shall we wait and see the results before you kill me?
KIRK: I don't intend to kill you unless I have to.
KOR: Sentimentality, mercy. The emotions of peace. Your weakness, Captain Kirk. The Klingon Empire shall win. Think of it, as we sit here, in space above us the destiny of the galaxy will be decided for the next ten thousand years. Can I offer you a drink? We can toast the victory of the Klingon fleet.>
SPOCK: You may be premature. There are many possibilities. >
KOR: Today we conquer. If some day we are defeated, well, war has its fortunes good and bad. Do you know why we are so strong? Because we are a unit. Each of us is part of the greater whole, always under surveillance. Even a commander like myself, always under surveillance, Captain. If you will note.
KIRK: Cover, Spock! Back!
(The Klingons burst it, then suddenly everyone drops their weapons. Everyone.)
The difference is that at least some of the Klingon recordings are watched in real time as they are made instead of merely being available for later study.
John M. Ford's novel The Final Reflection told the story of Klingon space navy officer Krenn and showed that naval security was always watching aboard a Klingon ship, and that the Klingon naval officers and crew feared and hated the rival Imperial Intelligence service even more than they did the naval Security service.