Translation:You are all busy.
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In the textbook 'Modern Mandarin Chinese Grammar' they are translating 很 hěn as 'very' in the sentence. 他们都很忙 Tāmen dōu hěn máng. They are all very busy. 她少她今天很忙. Tā shǎo tā jīntiān hěn máng. She said she is very busy today. But actually it should not be translated as 'very' according to this explanation:
The basic formula for Noun + “is” + Adjective in Chinese: Noun + 很 + Adjective.
This is a great guide for those struggling to known when to use shi and when to use hen. https://www.fluentu.com/blog/chinese/2015/07/09/shi-chinese/
Does this sentence need the 都 to be grammatically correct?
No, of course not!
Are both correct sentences?
Yes, but only the one with 都 is a correct translation for this sentence. If you don't include it, then it means "You are busy". Yes, you are still addressing a group, but you are not expressing the concept of "all".
很 is a bit of an odd character in that it does mean very but it is also most often used simply as what they called in my high school a 'descriptive particle', basically meaning that before an adjective, to indicate that it was as such, you'd place that character. I don't know if there is really an equivalent in English so I can't really compare it to anything but I guess you could think of it as a grammar equivalent of how you might place a dollar sign before a number to indicate it being a price, e.g. $9. So to indicate that 忙 is an adjective you place 很, the 'descriptive particle', before it to make 很忙. [NB: I'm only a learner myself so anyone feel free to correct or adjust my reply if at all incorrect! :) ]