There can be multiple answers on this question.
Official position of Ukraine is: "Embassy of Ukraine in the United States of America". Other official entities follow the same pattern.
Here's how "The Telegraph" puts it:
7. It’s not The Ukraine
The English-speaking world commonly referred to the country as The Ukraine. That is, until independence in 1991 when the West gradually dropped the definite article. In 1993 the Ukranian government requested that the country be called just Ukraine. US ambassador William Taylor, who knew that addition of the “the” was considered insulting by some Ukrainians, said it implied a disregard for the country’s sovereignty.
Average people in Ukraine. Unfortunately, there is no preference among Ukrainians. The main reason is probably due to the fact that many of us don't speak English well enough to see any difference on using the definite article with toponyms. Sad but true.
The level of foreign language competency in Russia is not better than in Ukraine (to say the least), so I don't think there are big enough communities there who would advocate using or omitting "the" when calling other states.
Neither Russian nor Ukrainian language has articles at all.
However, calling Ukraine by Russians has yet another notable issue.
In both Russian and Ukrainian languages, there are two ways to tell location: "in" and "on". According to grammar rules, "in" is used for state names, while "on" is for landmarks, territories, islands, etc. For example, "on Ural", "on the river(-side)", "on the North".
A vivid example: "on Cuba" means "island", while "in Cuba" means the name of the country. However, quite often they use "on" even in this case.
Russians tend to call countries that they have occupied or plan to occupy like they were lands, not countries. So the Russian government tends to say, "on Ukraine", not "in Ukraine".