In my post entitled “Thank you,” I made an annoying grammatical error. Did anyone catch it? I wrote, “[T]hank the person for their gift and don’t mention what you did with it.” See it?
The word their is a plural possessive pronoun, as in belonging to them. The word person is singular, and calls for a singular possessive pronoun, i.e. his or hers. Now do you see it?
I was once a barmaid in a pub in England (a long story for another time.) I had a gay manager who would sometimes confide in me his romantic woes. Since he wasn’t “out” to everyone in the pub, he would talk about his relationship partner using they and their rather than the person’s name. Since this was to protect his privacy, I consider it an excusable grammatical infraction.
But under normal circumstances, one should choose a side of the gender 
divide and stick with it when using possessive pronouns for an unknown person. I understand that gender in English is limited to masculine and feminine, and that there is no neutral ground. This being the case, I recommend to all speakers of English to come up with a plan for possessive pronouns and stick to it. Here are two suggestions:
1. Stick with (sexist) tradition and treat masculine as neutral (using his for all unknown persons) and feminine only for known females.
2. Let the sexes sort themselves, with men using him and his for all unknown persons, and women using her and hers.
One person remains one person and needs the singular. Work it out, people.
Shimshonit 
There is a trend (accepted by many grammarians and linguists) toward using “they” as the sexless third-person pronoun for things to which you do not wish to refer as “it.”
I wouldn’t use it in formal writing, however.
The use of “they” and “their” for the singular person pronoun can actually be traced back to the 19th century. At this point, it seems to becoming more and more an acceptable usage to counter the inherent sexism in using other pronouns for a non-specific gender.
Nomi and Michael: Thanks for the background on this practice. While it seems to have gained a measure of acceptance, and I know my schoolmarminess will have no effect whatever on its usage, I just had to say my piece. Agreement is a big pet peeve of mine, and while I do this no less than others, I try to edit it out before I click “publish.” It just bugs me.
I also deliberately use ‘their’ for gender neutrality. I regard myself as violating grammatical rules so that someday the rules will change – (religious analogy deleted).
It’s an annoying but sometimes necessary evil. Using plurals is a good way to dodge the issue, but it gets tiresome when you just want to say what you mean. I refuse to use the clunky “his/her” or the sore-thumb neologism “zie/zir”, although “(s)he” has a certain economy that appeals.
Consciously alternating male and female pronouns without a specific reason for doing so is overtly political and therefore distracting. I’d be fine with using “he/his,” but I’d rather be read for content than drawn into a defense of my supposed “chauvinism”.
It really bugs me to break the singular/plural conventions of the language further just so we can have a ‘gender-neutral’ way of speaking, but there don’t seem to be any other solutions.
Much as I love the English language, it has its limitations, and this is clearly one of them, for which there is no easy, non-political solution. Ah well.