I respectfully disagree that, "there was nobody there" is poor writing. To me, it sounds perfectly fine. I googled "there was nobody there," and found 176,000 links, many of which were written by professional writers. The first “there” in the sentence is a “dummy” (artificial) subject, used in English when the true subject has not been identified yet, but will be identified after the verb (usually the verb “to be”). Example: “There is a chair in the room.” The real subject is “chair,” which is introduced with the dummy subject “there.” The second “there” in the sentence is simply an adverb, meaning “in that place.”

Yes. It is definitely necessary. This is not a case of needless repetition. The two words 'there' have completely different functions within this sentence. The first 'There' is the subject of the sentence, a 'dummy pronoun' or 'expletive pronoun' that forms part of the expression 'There is'. The second 'there' indicates the location. Both are needed. And while we're analysing sentence structures with the verb 'to be', why don't you take a look at your question: "Does the second “there” in the sentence“there was nobody there” is unnecessary?" Hmm...not quite right, is it? I've noticed a school of thought here on italki along the lines of 'Don't bother constructing correct question forms. Just stick a 'Does' at the beginning of the sentence and a question mark at the end. Everyone will know it's a question and it'll be fine.' Go on, be brave. Have a go at asking the question properly.

It is definitely necessary. The second "there" refers to the place where nobody was

Just say "nobody was there". Unnecessary repetition of words does not qualify as good writing.

It makes sense, but it isn't needed. "He turned around and there was nobody there", doesn't sound as good as "He turned around and observed nobody was there." There are many other words, or ways to work your sentence, that are better than saying, there, twice.