Let’s say I become a citizen of a country that doesn’t allow dual citizenship. During naturalization, new country B tells me I have to renounce citizenship from old country A.
Does that have any effects back in country A? How would country A know? Would country A even care if they found out?
It varies from country to country. Some countries don’t let you become a citizen again after renouncing, while others allow it.
Citizenship is related to taxes (which also varies from country to country), so some countries are very interested in your citizenship in order to be able to establish that you owe them income taxes.
How would your previous country find out? I imagine it’s like any crime: you either do something to make it easier for them (try to renew a passport, fail to file a tax return) or they find you by accident (some investigator notices a connexion between two observations that makes their mind tingle).
There’s probably more, but that’s enough to answer your questions.