According to an article in NewKerala.com, the Veena Vadini school in Singrauli, India teaches its students to write with both hands, at the same time. And that's not all:

All these students are able to write simultaneously with both their hands. Trained from the early days at their school, these 72 young students are today at comfort with this rare art. They are also fluent in a number of languages.
Virangat Sharma, the principal of the school said that all his students are proficient in this art, which was started as an experiment. “The children are taught six languages Hindi, Urdu, English, Roman, Sanskrit and Arabic,” says Sharma. “I read somewhere that India's first President Dr. Rajendra Prasad used to write in two languages I also preferred to experiment developing such a skill among my students. All the children here can do this and also know the world's capital cities and their tables up to hundred. They can write on two different subjects and in two different languages at the same time,” says Sharma. Not just that these children can write with both their hands but they can also write in two different languages on two different subjects at the same time, tells Sharma.

Wow. And I thought my ability to write backwards in ancient Greek while doing a one-arm handstand and juggling two balls with my feet was impressive. Needless to say, I'm highly suspicious of the principal's claims. (Assuming that he exists and wasn't misquoted by a reporter.) The same story is also reported by ananova, adding to its credibility (note: sarcasm). I did a search for "Veena Vadini School" to see if they have a website, but only found links to this article about their instruction in ambidexterity. (Thanks to Kathy for sending me the clipping.)

Update:
SicTim (posting in the comments) remembered that Ripleys had once featured some cases of amazing ambidexterity. Checking the Best of Ripleys volume on my bookshelf, I found these examples. On the left, Lena Deeter of Conway, Arkansas, who "could write with both hands simultaneously — backwards, forwards, upside-down, even upside-down backwards! She could write in a different direction with each hand simultanously." (She appeared in a Ripleys cartoon on April 1, 1942... I assume she wasn't an April Fool's joke.) On the right is "a 1936 Dallas Odditorium performer [who] could draw three different cartoons simultaneously — with both hands and a foot!"

These cases indicate that it might be possible for someone to write in two different languages at the same time, but I'm still doubtful that an entire school could be trained to do it.