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.
I have said it before and I am sure I will say it again: In D&D 5e specific beats general.
The Rules
General rule on spell effects (PHB p 204):
Unless a spell has a perceptible effect, a creature might not know it was targeted by a spell at all. An effect like crackling lightning is obvious, but a more subtle effect, such as an attempt to read a creature’s thoughts, typically goes unnoticed, unless a spell says otherwise.
So, in general, if the spell has a perceptible effect that effect is perceived; if it isn't then it isn't. However, as always, specific beats general: if the spell says something different then that takes precedence.
General rules on hp & damage (PHB p 196)
Hit points represent a combination of physical and
mental durability, the will to live, and luck.
Whenever a creature takes damage, that damage is
subtracted from its hit points. The loss of hit points has
no effect on a creature’s capabilities until the creature
drops to 0 hit points.
hp do not exclusively represent the creatures physical body - the also represent metal ability, bloody-mindedness and luck.
General rules on psychic damage (PHB p 196)
Damage types have no rules of their own, but other rules, such as
damage resistance, rely on the types.
Psychic. Mental abilities such as a mind flayer’s
psionic blast deal psychic damage.
From these, there are no general rules on how psychic damage works.
Advice on describing damage (PHB p 197)
Describing the Effects of Damage
Dungeon Masters describe hit point loss in different ways.
When your current hit point total is half or more of your hit
point maximum, you typically show no signs of injury. When
you drop below half your hit point maximum, you show signs
of wear, such as cuts and bruises. An attack that reduces you
to 0 hit points strikes you directly, leaving a bleeding injury or
other trauma, or it simply knocks you unconscious.
Now, this is advice so how you want to describe it is up to you but according to the guideline: above half hp - no visible sings, below half hit points - visible signs, 0 hp - direct damage.
Specific rule for Vicious Mockery (PHB p 285)
You unleash a string of insults laced with subtle
enchantments at a creature you can see within range. If
the target can hear you ...
The specific rule is the enchantments are subtle, no one would notice these, possible including the victim (particularly if they save). For describing damage, in the absence of the specific rule you fall back on the general.
The Ruling
Whatever you want is fine so long as you consistently give the players the information they need.
Some people play in meta-game way with hp and treat them purely as numbers with no description applied - "You do 3hp of psychic damage; he now has 8hp left".
Others treat it as pure description with little to no feedback on mechanical effects - "You lash out with your insults, he pauses momentarily and blinks before raising his battle axe and snarling at you."
Or you can take the middle road - ""You lash out with your insults, doing 3 points of damage; he shows no obvious signs of distress."
Whatever works for your group is good.
TL;DR
If you want to follow the advice in the PHB then I would suggest the psychic damage will attack "mental durability, the will to live, and luck" manifesting as glazed eyes, trembling, tears, sweating, unprompted flinching etc. for less than half hp and bleeding from ears, eyes and nose for 0hp.
Best Answer
PHB p 204
So, yes, they remain unaware.