I have seen this question: "How does the amulet wearer control a Shield Guardian, and when does it act?" But it seems to fail to address the question of if casting a stored spell takes up an action, and if so, whose action: the amulet wearer's or the guardian's?
[RPG] Does the casting of a Shield Guardian’s stored spell take up an action
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The Shield Guardian is easier to understand if one doesn't go looking for details that aren't there, and if one notes that it's not a mindless construct. Treat it like a normal creature, with only the alterations made explicit in its description, and a lot of creature defaults will answer these points.
Commanding the Shield Guardian doesn't require a bonus action, because it doesn't say it does. To command the Shield Guardian requires nothing other than simply commanding it in the everyday sense of the word: saying words to tell it what to do. (Note in particular that its languages entry is “understands commands given in any language but can't speak”.)
It may be tempting to look at the Ranger's rules for commanding their animal companion, or similar examples where explicit actions are required: this is a trap, because the Guardian doesn't have any such wording. The actual normal case to compare this to is when monster A shouts commands to monster B, which doesn't have the limit of requiring a kind of action to do – they just do it whenever.
It doesn't say it can only act as commanded, and it does say that it has a focus on protecting the its master. It also has an Int of 7, so it's capable of simple reasoning at least, and probably more (7 isn't that far below average). Being unable to act unless commanded would be the unusual case that would need explicit statements, so default to normal creature handling: it does stuff on its own, following its own motivations (i.e., its single-minded focus on being a bodyguard), and also follows commands it hears from its master.
It acts on its own initiative count, because nothing indicates special handling regarding initiative. It has the capacity to make its own decisions (within its very narrow motives) absent commands, so it's not reliant on the amulet wearer's commands.
It doesn't say the stored spell is cast by the Guardian with “the same parameters”, it says “with any parameters set by the original caster”. Setting parameters is not required, then, it's at the original caster's discretion. This gives the original caster some optional extra control over the spell. The rest of the spell's parameters are up to the Shield Guardian when it casts it using normal casting procedures.
Commanding is done via words, not a special action that defines how the timing works. Words can be spoken during the amulet wearer's turn (or if your group has loosened the “on your turn” restriction, at any time), so the amulet wearer can issue commands any time they're able to speak. Either way, the Guardian will take action to follow its commands only when it takes its turn, as normal, since no extra wording indicates any deviation from the norm.
The Guardian isn't given a Bonus action or a Reaction by the Spell Storing trait, so it doesn't have one to use for this. It just does it as soon as it can, like any other order it follows or action it chooses itself. (It could use a Ready action to set up a Reaction with which to cast the spell, but that would just make it take longer to act after the situation arises, so it won't do that unless the situation is unusual in that the Guardian would get some warning and time to prepare.)
Yes, the Guardian concentrates.
The general rule is that concentration spells require the caster to concentrate, and there are no exceptions to this in the Shield Guardian’s description.
The fact that the Guardian has a low Intelligence score and has no general spellcasting ability doesn’t affect this — there’s no minimum ability scores for player characters to cast spells, for example. This is the same as if a low-Intelligence fighter were to cast a concentration spell from a ring of spell storing.
(In fact, the Guardian’s high Constitution score will make it quite good at concentrating. Although it is, surprisingly, not proficient with any saving throws).
“Parameters set by the caster” means any of the choices that would be made when the spell is cast — targets, modes if applicable for the spell, etc. In your example, targeting the rogue is a “parameter”. The fact that the spell requires concentration by the caster is just a property of greater invisibility, not a “parameter”.
Best Answer
While it's not specified, the action required to cast the stored spell should be the same as the action used to cast the spell normally, and must be taken by the shield guardian.
So the shield guardian is not just, for example, releasing the stored spell - they are casting it.
Not to sound obvious, but the casting time of a spell defines the time required to cast the spell. Since the guardian is casting the spell, they have to spend the appropriate amount of time.