How do you know if Americans genuinely/literally mean what they say?

In general, a genuine invitation is concrete, containing information that helps to make it happen. "Would you like to get lunch tomorrow?" is an invitation, and could be followed up with "Yes, how about [restaurant]?" or "Yes, do you have a place in mind?" to accept.

If you responded instead with "No, but we should meet some other time," that could be a polite refusal, but "No, how about Wednesday?" would indicate a genuine desire to meet.

Something like "You should visit us in DC some time" is not an invitation to visit DC, but might be an invitation to talk about visiting DC. So, at some point later you could call this person to discuss it. For example, "We talked about my visiting DC before. I was thinking of coming next month. If you'll be around it would be nice to see you." At this point, they'll hopefully make an actual invitation.

The point of all this dance is to avoid either party ever having to say, "No, I don't want to" or worse, "No, I don't like you". By forcing the issue immediately, you are putting people on the spot, forcing them to say "no" (or worse, "yes").

(I am no expert in US culture, but my own culture also has these non-invitations, and as far as I understand the meaning is the same in the US.)

To me, the key is the specificity of the invitation. The vaguest, of course, say "some time" - this is a bright light signalling that it is not a real invitation. "We should do lunch some time" means absolutely nothing at all. Similarly claims to "owe you" a beer or a coffee do not constitute an offer to deliver that item, nor an invitation to go, now or in the near future, to a place that sells that item. They are just things you say. It is possible the person who made that statement may later say "hey, let's go get that beer I owe you" which is both a genuine invitation and clarity that you're not paying. But until they do, the general statement of gratitude is not connected to an invitation.

In contrast, "We should have lunch if you come to my city" is a genuine invitation, subject to the person's availability when you happen to come to their city. You take them up on it by telling them "I'm coming to your city from the 11th to the 15th - any chance we can get together for that lunch?" They may well reply that they can't make it this time but maybe next time - and they might even mean it. However, statements about what "we should do" when you visit their city are not supposed to make you change your plans - if you reply immediately with "great idea, I can be there Dec 3rd" you are going to freak the person out a little. Instead, wait a few days, then say that you've decided to go to that city for your own reasons (that is, not on their account or because of what they said) and would love to include seeing them as part of your visit. The freaked out response is because they don't want to be responsible for you making an intercity visit, not because they don't want to have lunch with you. (I know, they just added an extra incentive for you to visit the city; thy should be flattered that you reply "that settles it, I am visiting that city right away!" but instead, they feel overly responsible for the entire visit, which is uncomfortable.)

The even more specific "we should have lunch next week" is a genuine invitation. You reply to that with "great idea, does Tuesday work for you?" or something similar. If you can't do it that week, but want to do it, you should have a substitute suggestion. Perhaps "I'm offsite all day every day next week and can't do lunches, but I could do breakfast if it was downtown, or dinner." Or "Next week is jammed solid, but can we lock down the following Monday so we can for sure do it then?" If you reply with a reason not to do it next week, and no specific substitute date, it will mean that you don't actually want to do it.

There are several factors that come into play here.

Some cultures have a significant amount of politeness as a social lubricant. Even when it doesn't mean you should, it is still said. A classic example of this is Japan's politeness (though this is simplifying a very large concept). There is an entire school of sociology called politeness theory.

Setting politeness aside, there is the different contexts that different cultures have. The wikipedia article goes a bit more into this at High- and low-context cultures. Other references on this High-context and Low-context Culture Styles from the College of Marin, and Context of Cultures: High and Low from University of the Pacific.

Most of the United States is a low context culture (the southern united states can be high(er) context). The example in Wikipedia is the stereotypical Texan saying lots with a prolonged silence while the New Yorker says something more precisely. The low context culture doesn't use as much understood social norms as part of the communication. People within a rural setting will likely be higher context than those in urban settings of the same culture. The stereotypical low context culture is that of the Germans and Swiss with the very precise wording and intent, while east asian cultures are the stereotypical high context cultures.

There is an art piece I saw a number of years ago that puts this into some other context for a Chinese born woman who lived in Germany from age 14 on titled East Meets West by Yang Liu (amazon). A number of these info graphics touch on the context differences between the cultures. For example, one on opinions - though I've also seen it written as 'complexity of self-expression' (Germany is blue, China is red):

Realize that this is meant as an art/infographic project meant to foster some amount of discussion and thought as to what it means and what the artist was trying to communicate.

With invitations, there are other factors at play here. The example given in the question of "we should have lunch some time" to "about next monday at 1pm" is a switch from the higher context social rituals to one that is now something more firm and commitment based. That transition can be jarring. The classic approach would probably be to give some time between the invitation (it is sincere) to the planning to reduce the switch from one type of communication pattern to another.

So yes, there is an intent to allow for a follow through on the invitation, though not right at that time.

In my experience, if an American says "you should visit me in DC", they mean it.

I'm an American myself, born and raised, and this has been my experience for over thirty years. However, I'm from rural New York, and now I live in rural California. In both of these areas, invitations like this are taken to be real. Considering how many upvotes the opposing answers have gotten, I'm wondering now if this is a city/rural cultural divide.

As an American, I have never managed to figure this out. To take some examples from the other answers, I've heard people say things like "You should visit us in DC some time" or "I'll owe you a beer" and later found out in some cases that it meant nothing, whereas in other cases they were completely serious.

So the only reasonable answer I can offer is that you can't know, at least not with certainty. It is true to some extent that, as the other answers are saying, a more specific offer is more likely to be genuine, but still you can't be sure.

On the bright side, many of the Americans who say things without meaning them also like to socialize. If someone makes you an insincere offer and you actually follow up on it, in many cases they will actually be interested. And if they really don't want to do it, you will be able to tell from their behavior: either they will tell you "no" directly, or they will repeatedly cancel or reschedule the plans.

To a greater extent than is true of other cultures, American expressions are "indicative" rather than "firm." That is that you should treat them as "tendencies," not promises.

If an American talks about getting together for lunch "sometime," s/he means a "random" time. So the correct response is not to suggest a definite time, but ask something like "when is a good time?" You might get a response like "sometime next year" in which case, you might say, "May I call you a year from now?" Or the response might be "oh, any time." In which case "May I call you in a year?" is still a good phrase because it's somewhat indefinite.

The American has expressed a preference for you over some other people, not an absolute desire to see you again. That means that you have "some chance" to see them again, but it's not a "sure thing." The American has reserved the right to prefer someone else over you at any given time.

It's confusing and, as an American, just as frustrating to me. People seem to want to look good, helpful, kind, caring, etc...when they're really not so much. They're essentially blowhards most often...braggarts, big and "gracious".

My advice? Wait for them to REALLY make a big deal out of it...embellish and get truly involved, instead of offering a rather off-hand comment. More details, more true enthusiasm, etc. Times, dates, places, etc. And I wouldn't accept "well, just call me..." That's likely to end in being put off.

I've only gotten real results when the conversation and comments continue and get more detailed and "enthusiastic". Off-hand comments are just that--their hot air blowing in the wind. And "sometime" is often really a euphemism for "probably never", I wasn't really serious, I just wanted to seem friendly and generous.

When hearing or saying this type of statement, I tend to view it in context and familiarity. In no case is a statement like this an actual invitation when spoken on its own. It is an analogous bidding of farewell to greetings like “How are you?”, or “What’s up?”.

In other words, if I make this sort of statement to someone I have just met, it is usually done because there is a reason to meet, visit, or get together with this person again. This reason could be to do business, exchange needed information, or to get to know the person better. It is an open invitation to an invitation upon which either party can act. This invitation means that I am open to either party making a real invitation in the future. Though, neither party is obligated to make or accept an invitation in the future. The future invitation is still one of high formality and should be treated/handled as if the initial invitation statement was not made.

However, if I make this sort of statement to a person whom I have a current or prior relationship (romantic, familial, business, acquaintance, etc), it is a genuine restatement of an already unspoken invitation. It is a reinforcement of the already established relationship. The statement does not need to be said, meant, or taken as being literal. It is just another way of saying goodbye. My openness to an invitation for another meeting, visit, or get together is already understood and a given. Unless I put special emphasis on the statement indicating the desire to meet again sooner rather than later, either party is open (but not obligated) to make an informal invitation at any time in the future.

If a statement like the ones mentioned is done in a context where the invitation is not only genuine, but also immediate, it will not be said as a stand alone statement. One does not make a statement like that as a genuine and immediate invitation (as opposed to an alternative to goodbye) when it is the final statement. “Let’s have lunch.” is different than, “Let’s have lunch. When are you free?” A genuine invitation will be followed by a question or inquiry, or a statement of options and plans. When these questions and statements do not follow the invitation, more emphasis and immediacy is placed on the statement specifically to draw out questions and statements from the other party. It is said as more of a command statement meant to act upon rather than a final statement to end the conversation. The affect is different to elicit a different effect.