I have researched and found out long time no see is a direct translate from the Chinese phrase, 好久不见. As much as we use it almost everyday in our lives, is it actually grammatically correct? Since it is a direct translate, is it grammatically approved?
Should we apply the terms grammatical and ungrammatical at all? That depends on whether English grammar has anything useful to say about it. And since it's a fixed-form idiom that doesn't combine with other words into valid English sentences, I'm not sure that is the case.
But I think we do parse it, or at least treat it as compositional. Why? Let's look at some evidence:
The phrase long time no X is a "snowclone", a phrasal template based on the original long time no see. Searching on COCA, I find several instantiations, all rare:
long time no hear
long time no hit
long time no interface
long time no speak
long time no talk
This is evidence that people do treat it as a compositional phrase to some extent, and not as a single word. Of course, these are somewhat rare, but the evidence is strong because they're all readily comprehensible. Compare understand with *undersit; the former is clearly not treated as compositional despite the surface resemblance to under and stand, and the latter is not readily comprehensible.
Long time is comprehensible, and an exchange like the following is plausible:
Alice: Hey, it's been a long time.
Bob: Yeah, long time no see!
In this hypothetical conversation, I think it's plausible that Bob's long time is a reference back to Alice's long time, so it seems to me that long time no see is compositional rather than opaque. It's equally plausible in reverse:
Alice: Hey, long time no see!
Bob: Yeah, it's been a long time.
So I think we do parse it as an idiomatic sequence of four words (meaning something like: it's been a long time, and we haven't seen each other.) And if we don't treat it as a single word, then we're putting the words together somehow, and that's part of grammar: how words fit together. So we probably can decide whether or not it's grammatical. So, is it?
Well, it's clearly ungrammatical by the rules of standard English. (And heck, it's ungrammatical by the rules of most non-standard varieties, too.) So we could apply that standard to it, but it doesn't seem particularly meaningful to do so. Idioms don't have to adhere to the grammatical rules that normal expressions do; some do, but many are special cases with their own rules.
In my opinion, if we must apply one label or the other, then it has to be grammatical! Why? Because we say it all the time! It may be informal, but it's perfectly standard. And sure, it's not grammatical by the rules of standard English, but those aren't the rules that apply to it. It has its own (extremely unproductive) rules. The facts that we understand derivative expressions and that we don't consider it opaque are decent evidence that this is the case.
But ultimately, it's not terribly useful to apply either label to it. What do we learn from doing so? Calling it grammatical is barely useful since it's a stretch to call it productive at all; and calling it ungrammatical is counterproductive because people will interpret that as a condemnation of a phrase that is standard English, and because we're applying rules to a phrase those rules don't apply to. So my preferred answer is this:
No, it's not grammatically correct, but it's not grammatically incorrect, either.
No. It's not grammatical in a simple academic sense of general English grammar.
However, many set phrases and common expressions, especially those with dialectal roots are widely accepted, understood and even considered 'standard'.
You may not be able to use the expression "Long time no see." in formal English writing, but it is fine in conversational English/ English chat, or casual writing.
The "direct translation" from Chinese is likely just a coincidence. Grammatical the sentence is not (as Yoda might say). It's simply an abbreviated (and easier) way of saying
"It's been a long time since last I've seen you." Or, "It's been a long time since I've seen you." Or, "It's been a long time since I saw you last."
There are probably other variations on the theme. It's nice, however, to have these little conversational shortcuts, which give us time to talk about more important things!
A more up-to-date abbreviated expression nowadays is
It’s a rather extreme form of ellipsis. The full version might be something like ‘It’s a long time since we’ve seen each other.’ Such ellipsis is also found in expressions like ‘More haste, less speed’. I wouldn’t say it was ungrammatical, but such constructions are perhaps best avoided in formal prose, other than for special effect.