Could the second "vi" in the phrase "vi pravas se vi pensas ke..." (you are right if you think that...) be omitted? Or is "vi pravas se pensas ke" defective?

No, not in formal speech. Perhaps in rapid talking in a crowded environment.

Rogir:No, not in formal speech. Perhaps in rapid talking in a crowded environment.

Thanks! That's what I suspected...

I tend to like the idea that one can ellipsize whenever one wants. I wouldn’t do this particular one in writing, but if I heard someone saying it, I wouldn’t correct them.

Would "Vi pravas se pensi ke" be understandable?

You have to be understood at least by every one, so don't be lazy when you speak

gyrus:Would "Vi pravas se pensi ke" be understandable?

See Rogir's message earlier. I would generally avoid short cuts like this unless you have seen them used by reputable authors. As you indicate, being understandable is the most important thing.

I really think that you should establish a good level of fluency before incorporating too much slang. It is important to draw the cultural distinction between Esperanto and other languages. In many languages, if you speak in an excessively "correct" manner, you sound stilted, foreign, and overly formal. In many languages, the correct use of slang and not-exactly-correct grammar is the key to sounding natural and native. For example, if I said in English "I know the person to whom you refer", it sounds formal and strange, although the grammar is perfect. "I know who you are talking about" is the colloquial use. Although ending a sentence in "about" would perhaps be frowned upon by grammarians, it wouldn't be possible to gain a "native sound" in English without the use of problematic grammar. Almost every language has instances like this, when you must break rules of grammar to sound natural and native.

Esperanto is a different story. Correct grammar is prized. Most of us are not native speakers, so using correct grammar with no mistakes is a sign that you have really "made it", you have really mastered the language. There are many people who speak Esperanto "fluently" but not correctly. They get their ideas across but frequently use the wrong words (often influenced by their native language) and get grammatical points wrong. They achieve the goal of basic communication but never invested the time in learning the finer grammatical points. This sort of sloppiness can lead to mistakes that can result in misunderstandings.

I understand that the people in this forum want to learn the REAL Esperanto, not just the "book language". I want to make sure that everyone understands that for the most part, "book" Esperanto is also the real spoken Esperanto. Speaking correct "book" Esperanto does not sound overly formal or pretentious in real-life spoken situations; it simply sounds correct (and people will be impressed that you have learned the language so well that there are no grammatical errors in your speech). Please understand that "colloquial" Esperanto is 99% based in slang words that are still correct Esperanto ("fresxbakita" = newbie), or else Esperanto cultural references (the many acronyms that dot the Esperanto landscape - UK, EK, LKK, etc), or grammar re-combined in unusual ways that are still correct (unusual word order to add emphasis, slangy words like "maltrinki"). Intentional use of incorrect grammar accounts for perhaps 1% of colloquial speech. This would include situations like dropping pronouns. I can't even think up too many of these situations off-hand, because in Esperanto, when you say things that are grammatically incorrect, you tend to sound as if you never bothered learning the language properly, and this isn't an image that most people try to project.

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Having said all that:

Beginners, do not try this at home. I do not advise that beginners experiment with this sort of poor grammar. Please note that all of my comments below are for information only.

In general, you do not hear omission of pronouns, even in rapid and informal speech. It can lead to too much confusion. But it depends a lot on the situation. If someone asks me, "Cxu Johano estas cxi tie?" (Is John here?), and I want to say "I don't think so", that would be "Mi pensas ke ne". But if someone kept asking me lots of questions, and the answer was always "Mi pensas ke ne", that might end up abbreviated as "Pensas ke ne". For other people the situation might be different but in my own speech, the only pronoun that ever gets omitted is "mi", only in extremely informal speech, and only in extremely short sentences expressing very basic meanings. In these cases, it's clear that I'm talking about "mi" since I've been asked a question and I'm answering it on my own behalf.

If anything, syllables tend to get dropped in rapid speech, not whole words. As you get deeper into the spoken language in real life situations, you get a sense of what gets dropped.

"estas" may become "stas". "Huuuu, 'stas tiom malvarme cxi tie!" (Oooh, it's so cold in here!)

The sample sentence was "vi pravas se vi pensas ke..."

Playing this over in my head, I can imagine it being pronounced rapidly as "Vi pravas s'vi pens's ke...". In very rapid speech, non-accented syllables could be de-emphasized so much that the vowel is nearly lost (that's how pensas could morph into something like pens-s)

How about omitting ĝi? It seems like a lot of places where we use "it" in English to refer to a previously mentioned concept, you could just omit the ĝi entirely. Is that ever "correct"?

No, it's not a choice, in certain situations you must not use it. That is in cases where 'it' does not refer to anything, and is used as a placeholder, for example

It was a bad idea to go there. - Estis malbona ideo iri tien.
It is raining. - Pluvas.
It was nice talking to you. Estis agrable paroli kun vi.

It is a white rabbit. - Ĝi estas blanka kuniklo.
It has been done. - Ĝi estas farita.

Well, I hope you get the idea.