The short answer is that depends.
My interpretation of the Army Language Policy contained in AR 600-20 paragraph 4-13 is that a Soldier should be required to speak English when performing official duties.
This issue can be a sensitive one. The Soldier has a responsibility to be proficient in English and the Leader has to ensure communications are clearly understood. Therefore the team must have an acceptable level of verbal communication. So if you are speaking about work related issues I would say English is a requirement. However depending on the situation the regulation leaves some latitude.
Notice the reference also states: “However, commanders may not require Soldiers to use English unless such use is clearly necessary and proper for the performance of military functions.” Accordingly,commanders may not require the use of English for personal communications that are unrelated to military function