This is a partial as I have access to...nothing at the moment for canon in-universe text to look at, so a bit speculative.
To whom are the calls directed?
Essentially to everybody (in reality everybody would have had basic training) but both sec chief/officer on duty and roving security teams (including closest) would know they are to respond.
This is partially due to shipboard internal communications (security request broadcast to all stations/locations).
In your example, all of those appear to be high priority calls for security. In that case, Chief of security at their station, their Number 2 or Officer on duty when the Chief is unavailable, comms and tactical officer, and, all security teams not just the nearest security team, would receive the call. From then on it is up to the security chain of command and the training of those members of the crew to decide what happens next.
Presumably there is sufficient staff to cover both the serious high priority security issues as well as to attend to the low priority ones ('My security pass doesn't work') - that we don't normally see on-screen.
So this answer is split in two - the first part more for answering the question, and then the second part is for tech that influenced that.
For security, presumably there is a particular station for ops/chief of security or whoever is on duty. But you would not rely on this as sole point of contact in an emergency.
Presumably during normal ops members of security would be making rounds anyway, and so a call on the mc would be received by everyone, but if its a call for security, only they know to react to it and the chef in the galley for example would be aware but can ignore it. That way the closest security personnel know they are closest and can attend without waiting to be told to do so.
This is faster than say routing a call to one duty station and hoping they re-route it to their officers on duty.
By doing continuous patrols the security team get to know every part of the ship they can get to without referring to the schematics/maps in an emergency, especially important on the larger ships as the series goes on. Apart from the chief at his duty station/office, security would never be in just one place at a time - they would be everywhere.
Security (and the ships system given it is the future) would have been trained to assess high priority and low priority and everything in between, so you would never get twenty of your nearest security officers coming to rescue a cat in a tree, but you would get those twenty to fend off an attempted boarding of the ship for example. Within security you would have rank and chain of command. Those at the top, like Worf, would not be dealing with trivial matters at all (unless he wanted to).
Kirk, in his physical position on the bridge and by his rank could use any of the other comms (1mc, 21mc, etc) at his disposal to call for security with the same response.
Also, in present day, you would have personal radio comms on you (com badges in Star Trek) and codes would be used that security would understand and know to respond to, that other members of the crew would hear but would know not to respond to nor to interfere in the radio chatter.
In terms of security personnel, this answer - https://scifi.stackexchange.com/a/71260/140676
refers to semi-canon listing of security, with 1 and presumably Number 2 (both Lt.Cmdr.), listed at the top.
Ultimately, Worf and Chekov (and the alternate universe Sulu) would be at the top of their security/ops department, made up of members of the ships company who have had extra weapons training and combat skills added to their resume.
So they do not get involved in really low priority security matters ('I lost my comm badge on shore leave') they would have a chain of command below them that allows those types of issues to get handled by others.
(The next bit is not really addressing the question but relates to the hardware involved)
Ships internal communications/influences:
From ottomh, it is not really addressed, but the theory is that as much as Gene wanted it to be far away from the military, systems such as ships comms would be designed along similar systems as naval ships of the present day:
communications cover several different types, and you might see all of them originating in a combat information center, which in Star Trek usually also means the bridge (or maybe the Battle Bridge):
- voice tubes
- ships service telephones
- pneumatic tubes
- multi-channel systems
- inter voice comm system (computer)
- combat information center comms group (specific stations within cic)
- sound-powered telephone systems
of all of the above in the naval ship, the multi channel system and the sound powered telephone seemed closest in on-screen operation in Star Trek TOS:
multi-channel systems were composed of two types:
intercom and ship wide announcement.
The former was composed of both a mic and an amplifier and had several basic controls, a bit like those boxes on the walls they hit to talk but to also hear. You can select which station to talk to (ie. Security) and the units can handle calls simultaneously.
The latter could be the 1mc or general announcing system and would be like Kirk broadcasting his mission to all of the ships crew.
Similarly, the Captains Command or 21mc provided two-way transmission of ship control orders and information among key stations only such as secondary battle stations (battle bridge), comms, damage control, engineering, tactical, etc. Both of these might be done from the bridge only.
I include sound powered telephone as i wonder if i saw the crew twisting a knob on a
box and then talking into it - this is the operation of a sound powered telephone, where a drum selector switch cuts in to any of the ship wide comm circuits thereby allowing a direct comm contact with a particular station (damage control, engineering, etc)
closest thing to a multi channel intercom in Star Trek TOS. (Hit the Red Button for ship-wide Red Alert)
US Navy mc intercom:
Shiny drum selector switch (red) to cut in on comms:
For an update to this, the TNG has the comm panel, as referenced in this answer:
PICARD: What is going on here, Number One? Did you give him permission to contact me?
RIKER: Of course not. He must have seen me use the comm panel.
In Enterprise they were called companels.
Additional Red alert buttons:
The panels also included a red button which sounded red alert when necessary. (TOS: "The Naked Time", "Balance of Terror", "Dagger of the Mind", et al.)
Trouble is, throughout the various series, it is pretty obvious that real-world knowledge of security operations, from high to low priority, is lacking in the writing, and does not have an in-universe explanation for that (MACOS in Enterprise was a nod in the right direction though), resulting in quite silly and weak security on the ships.
As an aside, most Navies put their crews through basic firearms training, so they can repel boarders or just know where the bang switch is, should the worst happen. Even auxiliary ships (replenishment/tankers, etc) might be outfitted with everything from small arms to cannons and the (picked) crew would need to be assigned and trained on them. In hostile waters is when the vessel takes on armed personnel that aren't normally part of the ships compliment - and the MACOS in Enterprise are an example of this, in-universe.
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