One is a lonely number

Nobody says that anything are possible, that everything are up for grabs, or that nothing matter. Yet almost nobody finds it jarring when somebody says that everybody has the right to their own opinion or that we don’t know someone until we live with them.
And yet, something is inconsistent here. Indefinite pronouns indisputably take a singular verb. Consequently, each of these indefinite pronouns also requires a singular pronoun (he/his, she/her, it/its) and not the plural they/their. "What happened?" is the title of a recent NYT article (3/14/10). What happened indeed that "In Washington, it's never safe to take at face value someone who swears they don't want a job in the White House”?
Despite each, every, either/neither, everything/everybody/ everyone, something/somebody/someone, nothing/nobody/no one, anything/anybody/anyone implying an indefinite number of people or things, these indefinite pronouns are grammatically singular. It is, therefore, incorrect to refer to "someone who swears" as "they."
R. W. Burchfield, author of The New Fowler’s Modern English Usage, believes that

All such ‘non-grammatical’ constructions arise either because the notion of plurality resides in many of the indefinite pronouns or because of the absence in English of a common-gender third person singular pronoun (as distinct from his used to mean “his or her” or the clumsy use of his or her itself (779).

Yet, if indefinite pronouns indeed carried a notion of plurality, then a writer would not use a singular verb in the first place but instead talk about someone who swear.
To say that everybody has the right to his or her own opinion is not necessarily clumsy and neither is "Everyone under 30 is comfortable revealing every facet of his or her life online" instead of "Everyone under 30 is comfortable revealing every facet of their lives online" ("A Tell-All Generation Learns to Keep Things Offline" NYT 5/9). In fact, his or her is in both cases appropriate since the sentences now clearly indicate that these facts and observations apply to both genders.
However, it can sometimes sound clumsy to use his or her to refer back to an indefinite pronoun; yet there are simple ways to avoid this without creating a “non-grammatical construction.”

In Washington, it's never safe to take at face value someone who swears they don't want a job in the White House (“What happened?” NYTmagazine 3/14/10).
Use plural antecedent: In Washington, it's never safe to take at face value all those (people) who swear they don’t want a job in the White House.
We don’t know someone until we live with them.
Rephrase: We don’t know someone until we live with this person.
If one wants their significant other to stay the night, they should run it by the roommates first (“Living as roommates without ruining the friendship” The Daily Aztec 5/10/10).
Eliminate the indefinite pronoun: Those/Roommates who want their significant other to stay the night, should run this idea by their roomies first.
Many people believe that the love one shares with their partner is enough to hold the relationship together (“How to stay together when thousands of miles apart” The Daily Aztec 5/10/10).
Here, the writer should simply stick with the original subject “Many people”: Many people believe that the love they share with their partner is enough to hold the relationship together.
While everybody should enjoy their experiences abroad, partners can remain loyal to each other by establishing reasonable boundaries (“How to stay...).
Rephrase: While everybody should enjoy the experiences a study abroad offers, partners can remain loyal to each other by establishing reasonable boundaries.
Don’t nag roommates: Everybody wants their independence (“Living as roommates...).
Rephrase: Don’t nag roommates. Just like you, they (= your roommates) want their independence.
Nobody should feel like a stranger in their own room or house (“Living as roommates...).
Alternate between she/he: Nobody should feel like a stranger in his own room or her own house.

Similarly, the word each is a pronoun that requires the singular because it refers to one of two or more things.
It helps when couples know what each other are going through (“Ways to work through depression together” The Daily Aztec 5/10/10).
Revised: It helps when each partner knows what the other is going through.

Finally, some sentences use they/their “because of the absence in English of a common-gender third person singular pronoun” (Burchfield). Again, this error can be avoided easily.
Encourage the depressed partner to see a therapist who can then recommend them to a psychiatrist if they need medical treatment. (“Ways to work through...).
Revised: Encourage the depressed partner to see a therapist who can then recommend a psychiatrist if medical treatment is needed.