Singular They – Can You Refer to Somebody Using They?

As found in Wikipedia, Singular They can be used for:

Indeterminate gender – when they refers to an individual person of unknown or unspecified sex, as in, for example, "One student failed their exam". This usage is known as epicene they.

Indeterminate number – when they has no definite antecedent, or can be interpreted as referring to either a singular or plural entity. This usage is also known as generic they. For example, in "Anyone who thinks they have been affected should contact their doctor", they and their are within the scope of the universal, distributive quantifier anyone,[1] and can be interpreted as referring to an unspecified individual or to people in general (notwithstanding the fact that "anyone" is strictly grammatically singular).

It further notes:

In some cases, they is used even when both the number and gender of the subject are known, but the identity of the person is generic, e.g. "If some guy beat me up, I'd leave them."

Though semantically singular or ambiguous, singular they remains morphologically and syntactically plural (e.g. it still takes plural forms of verbs).

To specifically answer the question posed:

Your reader is [...], but they are a busy person.

is correct. Even though your reader is singular, and when replaced by the singular they pronoun, they remains morphologically and syntactically plural and thus they are is appropriate.

You, of course, could bypass the whole issue, and say:

Your readers are [...]. but they are busy people.

Despite the fact that you're referring to a singular person, yes, the custom is always to maintain the same grammatical pluralization you would if you were using "they"/"them"/"their" in the more conventional sense. The SE phrasing, "Apparently, they prefer to keep an air of mystery about them", is therefore grammatical (including the last bit, for that matter), and the alternative is wrong enough in all cases that just about any native speaker would immediately notice and be jarred by it.

The only way to use singular verbs in such a case is to switch to a different pronoun. Using "it" to refer to people is extremely dehumanizing, so that leaves only two choices:

• the awkwardly slashed "s/he" and similar (including "he or she", hat tip Steve Jessop in comments)
• new words like "zhe", "ze", "zie", "zir", "hir", and so forth, which are relatively unfamiliar and confusing to anyone outside certain circles. (Especially because of the ridiculous number of ideas people have had for such new words.)

Usually, it's not worth the hassle. Just use "they"/"them"/"their" with plural verbs. No sane person will get offended, there's no grammatical problem, and stylistically the plural pronouns are arguably the least annoying in most cases.