Let's say a character cuts off one of an Ettin's heads, but the other head is still intact, and it's as such still alive. Obviously, there's a lot of blood. Does the Ettin take continuous damage from naturally bleeding out? Or does it not work that way?
The DM decides the effect
Although things like limb loss and hit locations aren't generally used in D&D, the Sword of Sharpness description provides a good edge case example: "you lop off one of the target’s limbs, with the effect of such loss determined by the GM".
For my own games: whenever a player reduces an opponent to 0 hp with a melee attack, I'm ok with describing the defeat however we want: dead, unconscious, disarmed, lost limb, etc.
No, it does not
For several reasons:
There are no called shots in 5e. Unless there is a mechanic for removing a head (e.g. the Hydra in the Monster Manual p. 190 or the Vorpal Sword from the DMG p. 209), there is no way to remove a head without houserules. Should a head be removed, there are mechanics in the stat block for that; remove a head and remove those mechanics.
The ettin has advantage on Wisdom (Perception) checks and on saving throws against being blinded, charmed, deafened, frightened, stunned, and knocked unconscious
When one of the ettin's heads is asleep, its other head is awake
The stat block of the Ettin provides no damage for loss of a head. Just loss of some mechanical advantages (see above.)
Unless you have a weapon or effect that specifically causes bleeding (e.g. Sword of Wounding from DMG p. 207), there are no ongoing bleeding effects from damage.