Generally when you ask to go out to eat 치맥 it's generally more common to say 먹자 with your friends & 먹읍시다 with your co-workers and boss. You really only utilize 식사합시다 with people that are older and your parents. It's more of a formality and situational usage. Hope this helps!
But it is, it's just that the subject is left out here. There are two words, the first is an object and tge second is a verb. (The verb is conjugated so it means "let's...", but still there's no subject)