RAW
Most of this is something that will have to be determined by your group / DM:
The rules don't clearly state whether or not the suggestion itself is the verbal component for Suggestion.
The rules do state that someone who makes a mental save is unaware of the spell unless the spell has a "perceptible effect" (PHB, page 204).
The rules do state that you need a material component for the spell, meaning you either need the component itself, a spell component pouch, an arcane focus, or a holy symbol.
But the rules don't really state how obvious it is when someone "uses" an arcane focus or holy symbol. Only that you must have a hand free to use a material component.
RAI
- Jeremy Crawford has stated that the Suggestion spell requires the chanting of "mystic words" in addition to the spoken suggestion.
A few Suggestions
Here are what I see as the best practices for this scenario.
If you want to take an action, but don't want players to know you're taking an action, you should make a stealth check.
See: How loud/obvious is a wizard casting a spell?.
This stealth check covers stilted arcane language, somatic gestures, glowing arcane foci, etc.
There would be a variety of skills to detect this
Insight — "Huh, he just said these weren't the droids they were looking for, and they repeated it back in a mono-tone? Weird."
Arcana — "That stress pattern. The eye contact. His hand in his component pouch. That's a Suggestion spell!"
Perception — "Why is this guy keeping that glowing green crystal hidden while he talks?"
Players should pick one, whichever is most meaningful to their character.
The victim of Suggestion is unaware of the spell unless given cause to question it.
Points 1 and 2 above apply to external observers. The spell wouldn't be much good if the victim could detect it.
I would not typically give the victim a chance to notice the spell, unless they made their save.
Q&A
Why stealth?
It doesn't have to be stealth. You could make a good case for deception, or sleight of hand instead. I picked stealth as a "catch all," and a skill that many casters would benefit from anyway.
Why would whispering the verbal component of suggestion require me to make a stealth check?
It isn't clear whether or not you can whisper your suggestion. Or if you need to speak it loudly. Or like Obi-Wan Kenobi in Star Wars. Or how obvious using a material component is (does it glow? does it need to be manipulated in the hands? etc.?).
If you like, you can define all of these aspects of the spell. If Suggestion comes up a lot, this might add quite a bit to your campaign.
But most of the time, I find it's just easier to let the dice decide. Rather than having a prolonged debate about the exact motions required to cast suggestion, or where everyone is standing, or how easy it would have been to notice a Jedi mind trick in the wild, it's more expedient to make a skill check.
Even if they discerned I was up to something, it would already be too late to discover what before the effect took hold.
None of this is intended to be a way to stop Suggestion. The spell should do what it says it does, regardless of what anyone sees or notices. This answer is about who is aware of what happened after the fact.
Wouldn't it require somebody else to make an active perception check to see if they could hear it? On top of that, they would also have to make an insight check to determine if they even knew what I was doing.
In general, if a character can make an active insight/perception roll to notice something, they should be able to use their passive scores without an action. You could have them roll against a DC if you want to emphasize the use of active skill checks (and even have different DCs for passive and active checks).
I generally prefer opposed rolls (or rolling against passive values), simply because it gives the player a bit more control, agency, and time in the spotlight.
No.
PHB, pg 203
If you can't provide one or more of a spell's components, you are unable to cast the spell.
When you cast Magic Mouth you are embedding a message in a usually mundane object to be spoken at another time. It is physically the object that speaks the words. Since you are not providing the spell's components, you are unable to cast the spell.
Best Answer
The perception rules give sample DCs. It is reasonable to equate "hearing someone speak in a strong voice" to "hear the details of a conversation" at DC 0. The DC increases by 1 for every 10 feet away, and a further +2 for the festivities (unfavorable conditions).
So if you try to dominate a guard who is patrolling 100 feet away from the nearest other guard, that nearest guard will need to succeed on a DC 12 check (+10 for 100 feet, +2 for background noise) to hear that something is going on. He would probably need to make an Intelligence check to realize that the noise was a spell being cast. If he doesn't know it was a spell, you could be long gone, and your new puppet can make up some reason like "Oh, I just stubbed my toe and cursed." Of course, the other guard would also be entitled to a Sense Motive check to determine traces of enchantment...
Note: By pure RAW, you do not need to actually speak in a strong voice, merely be able to, but you may have difficulty convincing your DM that it works this way.