Is this something people say when other people thank them, or when declining an offer? How would you say the other one?
To decline an offer: "Coffee?", "No, thank you" - Kawa? Nie, dziękuję
In response to "Thank you": "You are welcome" - Proszę/ Proszę bardzo
Dzięki or dziękuję? When do I use which?
"Dziękuję" is quite neutral, "dzięki" is informal. So you will rather use "dzięki" with friends only, or at least people you feel comfortable with. "Dziękuję" will sound ok even with them.
Just did a Danish. I almost put "Nie, tak".