I'm still quite young to this profession but I consider myself old fashioned when it comes to improper pronoun usage. The only time I accept the use of "their" over he or she is verbally, when mentioning the gender in question could compromise an identity that is supposed to remain anonymous.
To me, it is never acceptable in writing, and only minimally acceptable in speech.
Along the same lines, my blood boils when (especially common in interviews with professional athletes and entertainers) a person talks about his or her experience and continually speaks using the 2nd person "you" instead of simply saying "me," or "I." Example:
"How does it feel to be the most well loved female in pop culture?"
"Well, at first you feel a little overwhelmed. You're thinking, wow, this is a big responsibility. But then, you just sort of get used to it, and you start to understand that your talents and your fame can be used to change the world, you know?"
(Meanwhile, I'm screaming at my TV, "Me?! My talents or yours, lady?!")