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?
Yes, you lose all the rights you have as a citizen. That means you’ll need a passport and/or visa to visit if one is required, you can’t live there anymore without a formal agreement like a work visa, you don’t get any health insurance offered, you can’t get government assistance, you can’t vote, you can’t own firearms if that’s allowed, and any other rights that citizens in your home country have. You are no longer a citizen.
This all requires positively going through a process. It’s not automatic.
Yes, renouncing your citizenship is a formal process.
Not all countries require you to actually go through the process, some just have it in the oath.