how do logo designers know which fonts are fine to use and which they can't use for something public like a logo?

i.e. is every font in Illustrator free for public use? (the default ones ofc)

yes if you brought illustrator.

All fonts that are brought will come with a licence file informing you about what it can be used for.

The key phrase that you're looking for is "commercial use" of a font.

If you are unsure if a font you have can be used commercially, google it's license and check. Remember that trial versions do not grant you permission to use any part of the software for either private or commercial use, only to test the software.

If you bought the Creative Suite (or a singular program like Illu) then you have recieved a license to use the fonts (including commercial use).