I think I would translate it like "don't you want some water?" Because that is what feels natural to me. But I do hear some older people here in Oklahoma say "You not want some water?" But I can say that it is more rare for someone to say this than "You want water?" instead of "Do you want some water?"
I also thought "Want some chicken?" would be okay, and got it wrong. I feel like the 'do you' is normally implied, especially in informal situations. There is no 'Ты' or 'Тебя' here where we could infer the 'do you' from. The literal translation feels like 'Want (some) chicken?'.
We're at a point of conflict in idiom, it seems. In colloquial, very informal spoken English, "Want some chicken?" would be a natural thing to say, if asking a nearby friend or acquaintance in a relaxed setting. "Want chicken?" is strange in English idiom - it sounds incomplete, even in the most formal of settings.
If Курицу were Partitive/Genitive "Хочешь Курицы?", then "Want some chicken?" would be valid, wouldn't it?
Usually we don't say курятина. Купить курицу, готовить курицу, есть курицу (buy, cook, eat chicken).
My advice: if you see wings/chicken legs/other specific parts of the chicken, you should say курица. If you see only the meat (without specific parts that could let you know it's chicken), well, in this case you can say курятина.
— Это говядина или свинина (Is it beef or pork?)
— Нет, это курятина (No, it's chicken meat.)
I often suggest forvo.com as a good resource for hearing Russian words pronounced by native-speakers, but the two examples at https://forvo.com/word/%D0%BA%D1%83%D1%80%D0%B8%D1%86%D1%83/#ru make me less certain of that. The first one sounds really concise and accurate, while the second one sounds a bit slurred and sloppy - and pronounces "у" more like "o".
Could that be a regional difference?
(I received a mystery phone call from someone in Tennessee a few days ago, and although I could tell they were speaking some form of English, it was so mangled by the local accent or maybe dialect that I couldn't understand a word the speaker was uttering. And then there was the same experience with the doctor's receptionist from western North Carolina, except I got it after 4 tries.)
You're right, "Would you like some chicken?" is the polite way of saying this and would rather be translated into a way more polite Russian, like this "Bы хотите курицу?" or "Хотите курицу?". "Хочешь курицу?" is the same as "Ты хочешь курицу?" which is a little less polite.