"There are no people on the street."
Translation:На улице нет людей.
This sounds much less natural.
Russian usually places the new information in the end of the sentence. So, in «На у́лице нет люде́й» we tell information about what's on the street, and «нет люде́й» is the new information about the situation on the street.
In «Нет люде́й на у́лице», your listener knows there're no people (???), and you tell the new information about the absence of people: that people are absent on the street.
This works well only in one situation: when you have an enumeration, e.g. «нет люде́й на пло́щади, нет люде́й на у́лице, нет люде́й в па́рке» (there're no people in the square, there're no people on the street, there're no people in the part). I can't imagine another situation when «нет людей на улице» would sound good (with neutral intonation).
Also, please note that intonation can 'override' the word order. So, if you emphasise «нет людей» with intonation, then it will be understood as the new information. However, since we don't mark intonation in writing, such word order is not normally used in written texts.
Yes, it generally sounds weird. «Нет челове́к» might sometimes occur in colloquial speech, but «нет люде́й» is a much more common way of speaking.
«Челове́к» has an irregular plural form: «люди». Originally those were different words, but they came to be used as a singular/plural version of the same word. (The English ‘person’ and ‘people’ are slowly drifting in a similar direction.)
So, the genitive plural is «людей» in most cases. So, «нет люде́й».
However, there is one exception: if genitive plural is used after a numeral (i.e. after numerals ending in 5, 6 or larger), we still use «челове́к». So, «пять челове́к» ‘five people’.