They very clearly are not self aware. I regularly see them responding to their own statements as you/your (i.e. not having been generated by themselves). They do generally generate language in a manner consistent with the self awareness that we have and encode in our language, but that's the limit of it. Its the appearance of self awareness not actual self awareness.
Since LLMs have rudimentary internal models for concepts it is implausible that they cannot also have one for some simple sense of self. But a lot of people prefer the reductive explanation that the machine has never actually decoded anything at all from the language corpus.