English has no central authority, so the lines between grammar and style can get fuzzy. Most guides would say you need an "and" after the second comma, but some make an exception if the clauses are "very short and alike in form". See Wikipedia's article Comma splice if you are interested in the pedantics. https://en.wikipedia.org/wiki/Comma_splice

You still didn't answer: am I supposed to write "I'm eating a hot dog she's listening to radio" without a comma for that sentence to be grammatical? Or am I supposed to always insert "and" into it?

With or without a comma, without a conjunction like "and" the example is a run on sentence and is ungrammatical. You should join the clauses with a conjunction, and including a comma before the conjunction is a common style.

There's a caveat where some style guides permit using a comma without a conjunction in specific contexts, but most don't.