Quasi:
This is an interesting question for two reasons. First it touches on something the other posts did not touch: Does the other country tolerate dual citizenship? Just like it takes two to tango, it takes two countries to accept the idea of dual citizenship.
The US seems to turn a blind eye to it. They assume once you are a US citizen, you are a US citizen, period. What you do with your old passport appears to be your business. You may keep it as a memento.
But the other country might have different ideas. For instance, if you are a German citizen, living in the US, and your passport is expired, you have to declare that you are not a naturalized US citizen or have applied. And knowing the Germans, they probably check. They definitely check your Green Card, as if its their business.
So if you want to enjoy your “I zeee nozzink” type of dual citizenship for the maximum amount of time, you better apply for your US citizenship right after you have obtained your new German passport, because it lasts 10 years. After that, you either tell the truth or not, and I don’t know what happens in either case.
The second point is that in some countries, notably Germany and Israel, you never lose your right to citizenship if you fulfill certain requirements. If you are Jewish, you have a right to Jewish citizenship. If you are of German stock, you have the right to German citizenship. If your mother is German, you needn’t even be born in Germany. Hundreds and thousands of Russians have acquired German citizenship, simply because their forebears settled in Russia in the Middle Ages. On the other hand, a child of a foreign immigrant worker had no rights to German citizenship, even if he was born in Germany, even if his parents were born in Germany. They are changing this slowly, but reluctantly. Must be some of these old gene theories still floating around. So quasi, if you want to become German again, you march into the next German consulate or embassy, show them your birth certificate and other proof that German blood is in your veins - and you’ll be German again. I don’t know what happens with your US passport. Ask.