Should I ask my teacher before using speech-to-text software? Should I ask my teacher before using software of speech-to-text in classes ? The software in question does not record (the audio) at all, just "hear" the input and output text. 

What about using it in conferences and lectures ? And is it OK to share this with other students ? I'm more worried about legal issues than anything else (European Union and U.S jurisdictions).
Yes.

It is always polite to ask anyone for permission before acting towards them in a way that clearly differs from what is generally taken as the norm. 

I'm curious what you see as the argument against doing so, given you ask about etiquette.