Answer by Professional Japanese Teacher
Drinking is a big part in the Japanese corporate world. It is one of the ways to know your colleagues better and build camaraderie with them in an informal situation. There are basically two ways of saying this phrase.
Shall we go drinking?
Let’s go drinking
Let’s go drinking.
All phrases mean the same thing but the third phrase is the most casual one, only recommended for use with close friends. Do not say this phrase to your superior.
There is a slight difference in nuance between phrase one and phrase two. In phrase one ｎnomini ikimasenka, you are giving the listener an option whether he/she wants to go drinking with you or not.
However, in phrase two nomini ikimashoo, your volition is stronger and you are not giving the listener a choice.
So unless you are very certain that the listener will definitely go with you, try not to use nomini ikimashoo. Otherwise, the situation may turn awkward if the listener doesn’t want to go instead.