Is Lt Cmd. Giotto security chief throughout TOS?
And like are there multiple officers under him with rank of Lt. or are most of them enlisted/ensigns?
Also, what would their duties be?
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As with most of the original Star Fleet, the answer depends on what you consider to be "canon".
The sole mention of security leadership in the TV series is from TOS: Devil in the Dark when Lt Cmdr Giotto is detailed to assemble a security party. There's no specific mention of him being the Chief of Security and the official Startrek.com biography for him only refers to "a member of the U.S.S. Enterprise security team".
KIRK : Mister Spock, have Lieutenant Commander Giotto assemble the security troops and arm them with phaser number two.
Prior to this, the security teams merely seem to be assigned by whichever bridge officer needs them, at the time that they need them. There's no indication that any of them hold a rank higher than Crewman.
The Star Fleet Handbook (written by Geoffrey Mandel, and later incorporated into the semi-canon USS Enterprise Officer's Manual) lists Lt. Cmdr. Giotto as being "Head of Security" as of Stardate 1307 when Kirk assumed command of the Enterprise.
His team appears to consist of one other Lieutenant Commander and various Lieutenants. Presumably there are Crewman below them in rank also.
The revised USS Enterprise Officer's Manual (1980 edition, co-written with future TNG & DS9 Head of Art Doug Drexler) written in the wake of the success of Star Trek II indicates that Chekov took charge of Security following Giotto's retirement.
There is an interesting line in "Mirror, Mirror", when Scotty notes one of the differences he has observed between the barbaric Mirror universe and his own.
Mister Sulu is security chief, like the ancient gestapo!
The notion of a security chief is revolting to Scotty. Starfleet is an idealistic, radically peaceful organization, and this idealism is reflected in the noticeable tone of revulsion in Scotty's voice.
Others have observed that there is no named chief of security or security department in TOS. They don't exist because they are anathema to Starfleet at the time.