Verbal components are audible, but the message is not.
The message itself is separate from the verbal component
The message spell has a verbal component. However, the verbal component of the spell is not the same as the message itself, merely a necessary part of casting the the spell in the first place; the actual message is part of the spell effect. (Likewise, the somatic component of message is separate from the caster pointing at the targeted creature as part of the spell's effect.)
This is supported by a related official ruling in the Sage Advice Compendium regarding the spells suggestion and command, both of which also have verbal components:
Is the sentence of suggestion in the suggestion spell the verbal
component, or is the verbal component separate?
Verbal components are mystic words (PH, 203), not normal speech. The
spell’s suggestion is an intelligible utterance that is separate from
the verbal component. The command spell is the simplest example of
this principle. The utterance of the verbal component is separate
from, and precedes, any verbal utterance that would bring about the
All three of these spells have a verbal component, and separately involve speaking a word or sentence as part of the spell effect. They all work the same way in this regard; the verbal component that's part of the casting of the spell is distinct from the utterance spoken as part of the spell's effect.
Verbal components are audible, but it's up to the DM how audible
The description of verbal components in the rules states, in part (PHB, p. 203):
Most spells require the chanting of mystic words. The words themselves
aren't the source of the spell's power; rather, the particular
combination of sounds, with specific pitch and resonance, sets the
threads of magic in motion.
The rules don't specify how loud a verbal component has to be, so it is left to DM discretion. This is also reiterated by rules designer Jeremy Crawford in an unofficial tweet from October 2015:
when I cast a spell does the verbal component has to be clearly
audible or can I "whisper" the spell?
The verbal component of a spell must be audible to work. How loud
is audible? That's up to the DM.
The caster ostensibly doesn't have to scream it at the top of their lungs, but in general I think verbal components of spells would need to be spoken in (approximately) a regular speaking voice, or at a comparable volume.
DMs may choose to allow spellcasters to attempt to hide spell components (e.g. by whispering/muttering the verbal component) - potentially requiring an ability check to do so - but this runs the risk of devaluing the sorcerer's Subtle Spell Metamagic option (which removes verbal and somatic components from the spell entirely). Even beyond that, granting all casters a way to hide spell components for free generally makes all spellcasters more powerful regardless.
Only the target hears the message; only the caster hears the reply
As for the volume of the message, the description of the message cantrip states (emphasis mine):
You point your finger toward a creature within range and whisper a
message. The target (and only the target) hears the message and
can reply in a whisper that only you can hear.
The spell description makes it clear how loud the message itself is, and who can hear it. The caster whispers a message that only the target can hear, and the target can reply in a whisper that only the caster can hear. Thus, even if someone can tell that you're casting a spell, only the creature you point at can hear the actual message.