It is up to the DM, but the casting is noticeable
In 5e nowhere does it define anything specifically defining the volume of spells or how grand gestures have to be. These kinds of details are left entirely up to the DM to decide.
Most spells require the chanting of mystic words. [...] the particular combination of sounds, with specific pitch and resonance.
Spellcasting gestures might include a forceful gesticulation or an intricate set of gestures. [...]
This is all the detail we get from the rules. Clearly, gestures must be made and sounds must be produced, but the exact extent of beyond being audible and visible is not solidly layed out.
It is worth noting that the rules do even say that there is a potential range of how obvious the somatic component might be, so it may even depend on the spell.
Jeremy Crawford has shed some light on the intent for verbal components though (in an old unofficial tweet):
The verbal component of a spell must be audible to work. How loud is audible? That's up to the DM.
One thing is sure though: both the verbal and somatic components are definitely noticeable however. We know this because there is at least one class feature, a Sorcerer's Subtle Magic metamagic, whose only purpose is to make them not noticable:
When you cast a spell, you can spend 1 sorcery point to cast it without any somatic or verbal components.
In the end, your DM will have to decide how they want it to work in their world.