grogache

Russia and Ukraine are different in how they treat multiple citizenships, so talking about one doesn't really mean something about the other. (China is also different, in case the mod who closed the original thread is still confused.) It's true that a Russian citizen can gain citizenship in another country (at least in the US, in particular) without losing claim to Russian citizenship. However, a citizen of another country who wishes to gain Russian citizenship is required to renounce their former citizenship. In particular, you must write a letter to your consulate stating you wish to forfeit your citizenship. The actual effect that this will have varies depending on what the second country is. The US, for instance, requires that you renounce citizenship in person and thus a letter to that effect doesn't have legal force. An individual born in the US to at least one Russian parent (or an individual born in Russia to one US parent and one Russian parent) has a birthright claim to both Russian and US citizenship. Filing at the appropriate consulate or embassy would be required to obtain documentation to prove the second citizenship. The individual is not required to give up either of these citizenships at any time.
However, neither the US nor Russia allows dual citizenship. They allow multiple citizenships. Both countries treat you as either a citizen or not a citizen. If you have US citizenship, you are a US citizen. If you don't, you aren't. In neither case does it matter if you have Russian citizenship. The same is true of Russian citizenship.
As far as Russia and the US (but probably not Ukraine or China), the only time you would do something illegal is if you tried to use a foreign passport to enter the country that you are a citizen of. That is, if you are a US citizen, it is illegal to try to enter the US with a Russian passport. If you are a Russian, it is illegal to try to enter Russia with a US passport. In certain similar situations, it might be illegal to use a foreign passport when you have a domestic passport. Basically, while in Russia, you are Russian. Trying to use the US passport is illegal because you are basically denying being Russian when in fact you are. The same applies to trying to use the Russian passport in the US.
An individual with both Russian and US citizenship should use the Russian passport for entering and leaving Russia and the US passport for entering and leaving the US. Typically you will have to show your foreign passport to the airline worker to prove that you have documentation to enter the country to which you are flying. At least as far as Russia and the US, you wouldn't want to enter Russia with the Russian passport and leave with the US passport. For one thing, the stamps won't match up and for another you'll have trouble leaving with the US passport since it doesn't have a visa. I can see why that might work in Ukraine, but it's probably illegal.
On a side note, you can't legally apply for a Russian visa in the US passport or a US visa in the Russian passport since you shouldn't need a visa to enter the country you are a citizen of. This may raise the question why you would want to. Basically, it can come up if one of the passports has expired. Say you have been in the US for an extended period and the Russian passport has expired. The simplest course of action might be to get a Russian visa in the US passport, instead of renewing the Russian passport. However, since you are entitled to a Russian passport, applying for a visa is illegal.

Russia just doesn't acknowledge the other one if it doesn't have an agreement with the other country. So for the purposes of Russia, as far as i understand, if you only show the Russian passport they don't care that you're also a US citizen and treat you as if you were just a Russian citizen.
I wonder if it was the EU side that made her drop it then.

Two issues here with this , and the closed thread.
One, Russia allows dual citizenship and Ukraine currently does not or at least it seems that way.
Two, some mods like to flex their muscle and shut down threads to make themselves feel important.
I certainly did not see any TOS violation. It is a question that comes up often and needs to be discussed and understood.

EVERY single Russian/Eastern European native who has obtained US Citizenship has broached this topic! Essentially the native wants to obtain US Citizenship to make it easier to live with the spouse, to better assimilate to US life, or for any other reason, but also wants to keep the Russian passport so that it's simple to visit family and friends. My wife while she had her ten-year green card very easily renewed her external passport here at the Russian consulate. Also many many acquaintances and friends left the US on their US passports and arrived in Russia on their Russian passports. So: close this topic if you want MOD, but in doing so you're denying the forum members useful information and you're claiming "circumvention of governmental immigration law" when this "circumvention" happens all the time.
I'm going to write on the topic a bit, not to upset the MODs, but to show how (for Russia at least) this topic is a bit more multifaceted than simply an immigration law violation issue. I hope you all find it interesting!
This summer my wife will make her first trip home as a US citizen, and so we've asked every Russian/American we know how they handle this: apparently when arriving in Russia you just show your Russian passport, and when leaving you show the US one. Of course I'm shocked when I hear this since yes, Russia doesn't recognize dual citizenship, but I'm not shocked b/c of the draconian "circumvention of governmental immigration law" that the MOD reminded us of, but b/c I very practically understand that this situation presents a bribe-taking opportunity for the Russkies that is simpler than shooting fish in a barrel. (this is why in Moscow you always see eastern Asians being stopped by the police to check on their papers, but not Westerner expats who probably also are illegally working in the country - it's just much simpler to scare the former into paying up - there's a silly sitcom called Nasha Rasha, for those of you who speak Russian, that plays on some of these issues) For the OP or anyone else reading, I'll report back on how her trip went, and also I can tell you that multiple friends of ours made this trip in the fashion described and were never arrested/banned from the country/forced to pay bribes.
I think the reason that has never happened is due to the history behind Russia's rescindment of its dual-citizenship rules. Dual citizenship was just fine (and Russian citizenship for folks from FSU countries easy to obtain) initially following the Collapse of the USSR, which I guess made sense since, as Putin famously said, the collapse was "the greatest geopolitical catastrophe of the 20th century" that left ethnic Russians dispersed throughout the former USSR in what were now hostile-to-Russians countries. Russia wanted it to be easy for its citizens to come home fromthe FSU countries.
Anyway if you do a little reading you'll see that in 2002 Putin passed a law (for what purpose I don't know-I guess to satisfy his Nationalist supporters?) that greatly restricted citizenship rules and eliminated the idea of dual citizenship - FSU citizens had to give up their citizenship to become Russian citizens.
For those who are interested, you can also see the statement on the web site of the US consulate in St. Petersburg that sets up the idea of dual citizenship - the US basically says "you can keep your initial citizenship when you become a US citizen, but don't come crying to us if you travel and can't leave Russia to come back to the US b/c Russia doesn't recognize dual citizenship."