From what I have seen, most people communities learn all night on just the 1st night of Shavu'ot and not the 2nd night. (Perhaps, there are some that do both nights.)
If in Diaspora, we tend to treat the 2nd day of Yom Tov identically to the 1st, why don't we repeat the same idea on the 2nd night of Shavuot? True, that learning is a minhag (there is no mention of this as a mitzvah in the Torah).