In Western culture when people do shopping, they make a shopping list beforehand, or they use discount coupons.
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Is the expressions above grammatical?
Not quite. One says that people GO shopping, not DO shopping. The word "either" is usually coupled with "or", which implies that they either do one thing (make a list) or the other (use coupons) but not both. Actually, many shoppers do neither, and many do both. So the statement is not factually correct as it stands. It would be better to say that shoppers OFTEN do either or both.
Clipping coupons beforehand is equivalent to making part of a shopping list, but one may need to list other items as well.