Several answers have covered the catastrophic consequences of this power, both
in the short term and as we approach near-extinction. Let's talk about how a
society could stay meaningfully connected despite its new handicap, assuming it
doesn't destroy itself within days.
Children
@Frostfyre covered the problems that arise from infants/toddlers/teenagers being
able to kill with a thought. Assuming we can't save the ability until adulthood
(and that that would help), how do we survive our offspring?
The cop out
Maybe children are actually safe. Certainly in normal circumstances,
accidentally dropping a brick from a roof will kill someone as effectively as
intentionally doing it, but this power comes with the explicit definition that
you won't accidentally kill someone. If you don't yet understand that people can
end forever, can you intend to end them forever? However, even your child
remembering you from when they were 5 is a persistent danger.
Isolation
Infants can be nurtured safely until they develop coherent thoughts at all,
which gives them a year or two to be raised by their parent(s). Once they
develop any kind of intentional action or understanding of cause and effect,
they are moved to solitary confinement.
In isolation, interaction happens through modulated video and audio, and
necessary physical presence is done in a uniform that masks face and body (we do
this today to maintain
wildness
in rescued animals). Either we must trust caretakers, or the children are also
in uniform, and every interaction is randomized to avoid familiarity and
recognition. Education is automated and self-taught as much as possible. Of
course, this will have enormous detrimental effects on later social behavior and
mental health.
Socialization
At some point, you have to learn to be a part of society; we can't just bring
you out of a room after years in confinement and expect you to function. We
can't even do that properly with sane, convicted adults today. So as you grow
up, you get "friends" to the extent that it's safe. First, you learn to play a
game with a computer - a very mechanical game, with no real room for
improvisation or creativity. When you're consistent at it, you play it with
other children: in isolation over a terminal, then through glass, then
eventually in the same room. Both of you are in uniform, and if either of you
deviates from the game in a way that might identify somebody, the game is cut
short. The game gradually becomes more social until it involves multiple other
people and more freedom, but at that point you've all learned consistent-enough
behavioral skills that short interactions are feasible without leaving an
impression of individuality.
Graduation
Society needs some way of determining a person is fit to enter it, and passing
calculus is too low a bar. To be considered an adult and emerge from your
childhood confinement, you must pass rigorous psychological and behavioral
screening. Constant observation during adolescence provides history to a panel
that makes the decision. If we don't trust our caretakers, this observation is
anonymized - a pool of workers is given short clips of random, uniformed
children and asked to document their behavior. Graduates are allowed into
society on the basis that their education ideally has prepared them to control
their impulses, but at least taught them to keep themselves indistinguishable.
What happens to those deemed unfit to enter society is left as an exercise for
the reader.
Social norms
On the assumption that you trust no one, or very few people, social interactions
hinge on conducting oneself anonymously. The most obvious aspect of that is a
uniform that masks your face, body shape, gait, voice, any anything else we're
concerned about. I'm thinking
Rorschach jumpsuits and
voice modulators. Maybe people express themselves through fashion, but if they
do, that season's color is truly ubiquitous. The morning news tells you what the
most people are wearing that day, and on Friday we all wear Hawaiian shirts.
You don't have a name, or if you do, it lasts only as long as the current
interaction. Conversation is terse and communicates only the necessary
information, with no displayed emotion or identifying idiosynchracies. Come to
think of it, when this calamity strikes, hope you're on Vulcan. People with
identifying characteristics like tics, speech impediments, or unmaskable
physical differences leave their home at great risk.
You don't tell anyone where you live. You move frequently so your neighbors
don't notice your habits, which you do your best to randomize. Either you have a
dozen different soaps so you smell different every day, or everyone uses the
same generic brand.
Technology
A plethora of automated assistants appear that help you manage yourself. They
alert you if you use a phrase unusually often or visit the same coffeeshop
too regularly. Everybody uses them, and no one talks about them, because they
warn you not to mention what they warn you about.
All browsers block ads and use Tor, because a database of identifying marketing
information is a database of death wishes. Either Facebook shutters itself on
ethical grounds, or the rush to delete profiles DDoSes the site into oblivion.
Your cell phone is paid for in cash or cryptocurrency (credit cards collapsed
right along with the marketing industry) and randomizes its IMEI constantly.
Only data plans remain, since it's not like you're giving out your number. Soon,
no one will remember why it's called a "phone".
Law
Luckily, this dystopia doesn't need imposingly anonymous, masked police.
Everyone's anonymous and masked, the police just carry guns. Most crime is
unchanged, since "the masked man who stole my purse" isn't enough to kill,
though identity theft is a cute euphemism for assassination.
Judicial proceedings are simplified because it's virtually a requirement that
the defendant be arrested during or just after the crime is committed -
otherwise, they have no identifying characteristics. They are kept in isolation
both to protect them from the plaintiff and just to keep track of them until the
trial, which consists of each person privately telling a judge their side. No
one's entitled to a jury of their peers, and no one wants a dozen strangers
getting to know them anyway. Fines must be assessed immediately. Prisons are
100% solitary confinement. Capital punishment amounts to publishing your face.
Work
Everyone works on a team of people chosen at random from a pool representing
their skill set. Coworkers rotate through constantly, because staying amongst
the same people for any amount of time is dangerous. Everyone has at least two
co-bosses, who are rotated similarly, and the redundant structure goes all the
way to the top. This provides enough continuity for work to move forward without
anyone being able to distinguish an individual, but there is efficiency lost in
the constant churn.
Employee evaluation is pointless, since there is no lasting record or means to
recommend you to future employers. Workplace issues like safety or harassment
result in immediate termination, because there is no way of giving you three
strikes. You get paid at the end of each day because there's no telling if
you'll come in tomorrow. If you misrepresent your ability to do the job, they
call the cops.
Relationships
It's obviously difficult to develop friendships, let alone romantic involvement,
though there are plenty of meetups for like-minded people in sufficient numbers.
Making a real friend is one of the most difficult, dangerous, intimate things
you can do. Everyone is constantly on guard and just passing through, but there
is enough interaction that you'll eventually find just a small handful of people
to very, very, very slowly get to know. Since everyone's passed their
ready-for-society test, there's actually not as bad a trust baseline as their
could be.
A sufficiently close relationship—which perhaps we could call marriage, though
it doesn't have to be—is bonded by showing your faces to each other in the
presence of a (safely masked) third party officiant. The officiant is employed
by the state (or some other trusted entity), and gets a feed of recently
deceased faces, which he checks every day for one of you. If one dies, he kills
the other.
The officiant makes long-term relationships safe, sexual or not, but obviously
that's a supremely high bar. All other sexual encounters are like the rest of
society's interactions - brief, impersonal, and anonymous. Undressing is risky
if you have an identifying birthmark, but alcohol helps erase the details.
Seeing someone's face is the obvious fetish, and there's a whole industry
around CGI heads and human masks.
Optimism
That this society provides no room for error is stressful, but has its upside.
Concern for their life if they tick someone off keeps them civil, maybe even
kind. Pre-killing-power, calling someone a name might get you yelled at, but if
you're not willing to risk death on the other person not figuring you out,
you'll hold your tongue.
And for the most part, nobody wakes up wanting to kill people. Just as you're
trying to avoid doing anything that'll get you noticed, you actively ignore
everyone else so you don't accidentally gain the ability to end them. Everyone
generally minds their own business and avoids conflict.
Pessimism
This society lives in constant fear of retribution for the slightest social
misstep. They wear masks outside, try to look and act like everyone else, and
don't open up to anyone in case it comes back to haunt them. Everybody exists in
their own, isolated universe, passing by myriad other equally inscrutable,
equally lonely people, always wondering if they can risk getting close or
trusting someone.
But really, does that sound so different?