The basic definition of native that applies here would be:
3) related to one as, or in connection with, the place of one's birth or origin" ⇒ one's native land, one's native language"
So as I was born in America, I could say I am a "native American".
As for language, for example my parents were German immigrants, but being in America only speaking English in our house was allowed. As children, even when all the relatives got together for some holiday, us kids had to sit at the kids table, so the adults could discuss stuff away from us (in German).
So I am a "native speaker of the English language". My parents would be "native speakers of the German language". It does not address where we are.
As for your example:
It's ok to start speaking in [Hindi/Maori/Mongolian/English/Japanese] with the natives from day one.
Here native means persons born in the country you are referring to (you could substitute local people for natives without changing the meaning).
Unless you are on an Pacific Island where nobody ever arrives or leaves, native location and language may not be the same.
Also the word natives (used by itself) has been used a lot (especially by Hollywood) to describe primitive peoples in a fictional setting (such as a jungle), who are usually cast as inferior to the (usually white) educated protagonists. It is possible some people might be offended.