When wanting to work in the UAE, how important is it to be fluent in English and other European languages?

If you looked around, it's not just in UAE , everywhere now demands the knowledge and fluency in English. Everybody must know English nowadays.

The most important is to be fluent in English, employers will ask it in every vacancy. Second most important are arabic language skills. Other European languages like German and French are third most important. I have no clue for the rest for the languages, but it is always an extra add to your skills.

Has english is world wide know languages. So its the basis requirement to know and understand the language.

Very Important at least to be good in English almost is the main language specialy in Dubai.

wherever you are English is must, the second option is Arabic if you know these two languages you don't have to worry about anything in GCC.

English is a dominant language all around the world therefore the more fluent one is  in english the more communicable you are in terms of relating to everyone. Being also fluent in other languages are very important because communicating to a wide cross section and diverse group of people makes you stand out.