Hello,
Did you make sure to start on the exact same quests? If your teammate is doing quests that you don't have unlocked yet, they will not count towards your quests.
Also, if you've been playing for less than seven days, I suggest you take a quick look at this article.
https://www.bungie.net/Mobi/en/About/Article?id=13962
English
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I have had this issue as well with the xur weekly exotic cipher mission. It doesn't count if I'm not lead? Seems a bit obsurd to me..but okay
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Yes, we all started from the beginning together. And talked to our respective class leaders after every quest. I noticed half way through that my quest was not progressing along with my Fireteam leader and then it required that I start back from the first mission.
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Can you and your teammate both try clearing your caches? https://www.bungie.net/Mobi/en/About/Article?id=12539 It may help.
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That's very odd, I will ask about it.