OK, I will have to accept that decision, and thanks for your reply. But for the sake of argument I will point out that if you replace 'big' with 'huge', it works in English. For example, it is standard English to say 'I have a huge thirst for knowledge' or even (less metaphorically), 'By the end of the day, I had developed a huge thirst for a good beer/lemonade etc'. So I think that the expression 'a huge thirst' is quite common in English, even if 'a big thirst' isn't - and after all, 'big' and 'large' are synonyms in English. A lot depends on context, and because English has so many synonyms, what sounds awkward with one word can often be common with a synonym. So I'm thinking that if someone answered 'We have a huge thirst' it ought to be acceptable, even though it seems that 'We are very thirsty.' would be the closest English equivalent to the Czech idiom.
"Jsme velmi žízniví" (the literal translation from English) actually sounds more like it's our trait - i.e. we are very thirsty people, all the time, not just now. It could also be used to express "We are thirsty (now)", but it's quite uncommon - for instance, someone might use it on purpose to sound funny.
"Jsme (velmi) hladoví" is used a little more frequently, but again, the standard expression is "Máme (velký) hlad".