Even though the player character in Dark Souls can be in a "hollow" state, the player never truly goes hollow. At least, not in the sense that NPCs and enemies have gone hollow.

According to Dark Souls' mythology, the undead are condemned to repeatedly wander Lordran in search of a cure, being unable to permanently die. But for virtually all such undead, this quest is futile. An undead can temporarily stave off hallowing by absorbing souls or infusing themselves with the humanity of someone else. Eventually, an undead dies one too many times, or is worn down by the daily grind of collecting souls, and loses the will to go on -- or is simply unable to continue collecting souls and humanity. When this happens, that undead becomes hollow, loses his sanity and free will, and continues to wander the world as a mindless zombie attacking any un-hollowed that it encounters on sight.

It is unknown how many "Chosen Undead" are brought to Lordran or the Undead Asyulm, but the Crestfallen Warrior at Firelink tells us that many have come before you. Is it possible that all hollows in Lordran were at some point "Chosen Undead", tasked by Frampt to retrieve the Lordvessel and re-kindle the dying flame?

Probably not.

A great deal of the hollows that you encounter in the game were likely former residents of Lordran, and there was no need to select a "Chosen Undead" until Gwyn's power faded to a "cinder", and the fire began to die. This presumably took a very long time - a whole "age".

The Crestfallen Warrior informs us that we are not the first "chosen undead",
and suspects that we won't be the last either.

Avoiding hollowness with purpose

Many undead adventurers wandered into Lordran (or were abducted and taken there), and they struggle to hold onto their precious humanity for as long as possible, fighting for their lives in the fear that they, too will go hollow. Some, like the Crestfallen Warrior, resign themselves to the inevitability of hollowness, and find a sense of purpose in warning other new arrivals that they, too, are doomed. Others pursue some seemingly impossible goal or objective in the hopes that the journey will provide them with the sense of purpose necessary to avoid (or at least delay) hollowing. And yet others have taken up crafts or vocations such as blacksmithing, vending, or guarding something in order to keep them focused and avoid hollowing (and to exchange goods or services for the very souls that they need to stave off the hollowing). Keeping such a goal may help keep an undead partially lucid, but they also seem to begin to forget everything else, and only the knowledge of their quest or craft remains. Perhaps, the undead guarding various areas of the game were, at one point, tasked with protecting that place (or something within that place), but have long since lost their mind, and only that compulsion to defend has remained.

Going on "one final quest" seems to provide adventurers with enough focus to hold back hollowing.

But hollowing isn't just a thematic element reserved for non-player characters; hollowing is also a mechanic in the game that affects the player. Whenever the player character dies, you are reborn at the last bonfire in a hollowed state, unable to summon help from allies until you restore your humanity through the consumption of someone else's humanity. In Dark Souls II, hollowing further handicaps the player by cummulatively reducing your total health each time you die, and only restoring your humanity can refill your health meter. In both these cases, the player is not truly hollow; you are only in a state of partial hollowing.

It's unclear whether non-player characters are able to die and restore their humanity, or if deaths contribute to an irreversible progression towards hollowness. There are, after all, apparently hollowed NPCs such as the undead merchant in the Undead Burg and blacksmith Lenigrast in Majula who are sane enough to have kept their shops open. The presence of NPC summon signs hints at the possibility that they, too, are capable of restoring their own humanity through the same mechanisms that you can, but the game itself justifies this with ambiguous appeals to "time distortion" and hypothetical parallel realities that obfuscates the matter - particularly where Solaire and Lautrec are concerned.

Summoned NPCs may recover humanity as you do, or they're from another time or dimension, or both.

Solaire's dialogue refers to "heroes centuries old phasing in and out.". Solaire may be using the words "world" and "time" interchangeably. This seems to be the game's justification for how summoning works: you may be literally summoning someone from a bygone era into your own time period. Anytime, you are summoned to someone else's world, you are also being transported to another time (past or future, depending on whether or not you finish the game). Solaire and Lautrec seem to somehow come from another time or dimension, but other characters definitely seem to exist within your world and time: Andre, the Crestfallen Warrior, Rhea and her companions, Big Hat Logan and his apprentice, and so on are all undead who have seen many other "Chosen Undead" come to Lordran seeking their destiny.

In any case, it's not until an undead "gives up" that the hollowing process becomes complete. What do we mean by "giving up"? For an NPC, it means that they gave up on life and went hollow, and the player typically ends up putting them down. For the player, it means that you stop playing the game. As long as you continue to play the game, then your character will continue to hold onto a sliver of humanity and maintain his or her sanity for a little while longer. When you put down your controller for the last time, you have condemned your character avatar to finally succumbing to hollowness, whether you recognize it or not...

UPDATE SEPTEMBER 1, 2016:
The release of Dark Souls III has finally answered this question. That game has rendered the speculation below completely moot and invalid. However, this post will remain preserved for posterity.

There is considerable debate within the Dark Souls fan base regarding whether or not Knight Solaire of Astora may be the forgotten son of Lord Gwyn, and the God of War. The game contains many references to a forgotten God of War, who was the son of Lord Gwyn, the God of Sunlight. The primary source for this is the item description for the Ring of the Sun's Firstborn:

"Lord Gwyn's firstborn, who inherited the
sunlight, once wore this ancient ring.
Boosts the strength of miracles.
 
Lord Gwyn's firstborn was a god of war,
but his foolishness led to a loss of the
annals, and rescinding of his deific status.
Today, even his name is not known."

The game doesn't specify what the God of War did to be stripped from the annals, but the blunder cost him dearly. As a punishment, Gwyn and the other gods rescinded the God of War's diefic status and expelled him from Anor Londo.

But it didn't end there. Based on the content of the game itself, it appears that the gods also removed all references to him. This included removing or destroying any statues depicting him and redacting his name from records. Both his name and face are lost to history, as is his fate. Was he cast out of Anor Londo? Was he made mortal? Was he cast out of the world, entirely?

The gods made an effort to eradicate all records and traces of the God of War.
Statues of him were removed or destroyed all throughout Anor Londo and Lordran.

If the Warriors of Sunlight covenant (lead by the God of War) existed prior to the God of War's expulsion from Anor Londo, then it would also stand to reason that the gods disbanded the covenant. It's highly unlikely that gods would have permitted the covenant to continue to function, since its followers would be able to continue to teach of the existence of the God of War. These followers would likely have been forced to renounce the covenant or become an underground cult. The fact that the covenant still exists and has followers is likely due to the waning influence of the gods after Gwyn left to link the flames and the gods were forced to abandon Anor Londo.

The Sunlight Medal does offer one clue as to the fate of the God of War and his covenant. It suggests that the forgotten god still lives, and still watches over his followers:

"This faintly warm medal engraved with the
symbol of the Sun, is the ultimate honor,
awarded to those who summon the Warrior
of Sunlight and complete a goal.
 
The symbol represents Lord Gwyn's firstborn,
who lost his deity status and was expunged
from the annals. But the old God of War
still watches closely over his warriors."

In any case, we meet one NPC Warrior of Sunlight during the game: Solaire of Astora.

Solaire doesn't reveal much about himself. He is a Warrior of the Sunlight who has a strong reverence for the sun itself. In fact, he is on a possibly futile attempt to obtain his own "sun"...

I recently posted my much-belated review of Dark Souls II. In it, I criticized the game for having lackluster online components, but didn't go into much detail other than to say that Soul Memory seems like a non-optimal matchmaking method and that invasions are rare and reserved for elite players. I wanted to take a moment to go over some of the other complaints that I have with the game's online mechanics, as well as to offer some suggestions for improving them. While it seems unlikely that From will make significant mechanical changes at the fundamental levels that I am about to propose, the fact that there is still at least one more DLC incoming means they have the opportunity to do so.

Human Effigies were too rare to be
the only means of revival.

When the game initially released, humanity could only be restored by consuming a Human Effigy. This mechanic was an interesting departure, since the previous games had both relied on defeating bosses as the primary way of reviving. The idea of requiring a consumable item to restore humanity wasn't exactly earth-shattering or fundamentally broken, but the specific implementation had one major flaw: Human Effigies were very rare, and there was no way to farm them!

This made deaths feel extremely punitive and proved unpopular with players and critics, and so From reversed their design and went back to granting revivals from boss kills.

After the 1.03 patch, maintaining your humanity has become almost trivial. This is mostly due to the fact that the White Soapstone can be placed anywhere in the level (including right outside a boss's fog gate), and that players can use the Small Soapstone to fully restore humanity by spending a couple minutes killing standard enemies. By using the Small Soapstone, you spend a few minutes in another player's world, and killing enemies shortens this duration. At the end, you are sent back to your own world with fully restored health, humanity, item condition, estus, and a small reward (usually a Smooth & Silky Stone).

Revival via the Small Soapstone makes it far too easy to keep your humanity. It requires only a minimal investment of time and effort, and you don't even have to beat a boss to fulfill it. This practically nullifies the cumulative loss of health from hollowing and makes it almost trivial to maintain humanity throughout most of the game.

The Small White Soapstone can be used to easily restore full humanity - sometimes without any effort by the player!
Instead of full restoration, it should provide a partial restoration.

Fixing this imbalance seems easy enough: completing the Small Soapstone task should only partially restore the player's humanity. Instead of a full restore, the player's max HP could be restored equivalent to a single death...