It is clear from the answer to this question that all astro- or cosmonauts need to study English or Russian, whichever is not their native tongue. But as a practical matter, what language prevails for daily operations when two people on the ISS have to communicate, and are not from the same country? I was especially struck by this video the two guys from the one-year crew are answering questions from a NASA interviewer, each in their native languages.
There is a film "Русский в космосе" (Russian language in space, 12.04.2014) by Russia24:
They say that Russian language is popular, and (18 minutes in) an unofficial mix of Russian and English is used - рунглиш (Runglish) or русам (RusAm). They often use native language of the person with whom they want to talk (this rule was introduced during the Apollo–Soyuz test docking).
For example in this video, Sunita Williams says "извините.. это пробка" in Russian ("sorry, this is traffic jam") to a Russian cosmonaut and he replies "no problem" in English.
The paper Does Russian English Exist? (American Journal of Educational Research, 2014 2 (9), pp 832-839. DOI: 10.12691/education-2-9-20) says:
Russian English (RE) has already attracted public attention. The term 'Runglish' was coined in 2000 for one of the languages aboard the International Space Station, a mixture of English and Russian [Feuer, A. For the Thirsty Runglish Speaker: Try an Ized Ciawfeh. The N.Y. Times 14 June 2005.].
And a newspaper from 2000:
The crew said they will rely on a mixture of languages and on a mixed cuisine when they are aboard the international station.
"We say jokingly that we communicate in 'Runglish,' a mixture of Russian and English languages, so that when we are short of words in one language we can use the other, because all the crew members speak both languages well," Krikalyov said.
"The menu will be 'Runglish' too: partly American food and partly Russian," Shepherd added.
English Wikipedia also has a short article on Runglish. In Runglish, if you don't know some word in the current language, you can say it in another language:
The term itself, in any case, is commonly traced to 2000, when the not-quite-bilingual Russian-American crew of the International Space Station coined it to describe their on-board speech: Lacking a word or phrase, they used what they knew and filled in around it ("Давай маленький Phillips screwdriver, Костя" -- give the small Phillips screwdriver, Kostya).
From another ISS crew interview from 2013:
А на каком языке вы будете общаться?
(What language are you going to communicate in?)
Today we are going to use Runglish. This is our unofficial language of the ISS program. It is called Runglish, it is a blend of English and Russian.
1$\begingroup$ There is a writings on the wall, shown in nov 2011 tour youtu.be/3ErLtE3Lf9s?t=63 "Не Трогай этого рака (a3 rack)" = do not touch this rack, the russian word "рак" (literally Crustacean) was used to describe word "rack", possibly unknown for the writer ("стойка") due to similar pronunciation. There is other label near - "Do NOT TOUCH this A3 rack" $\endgroup$– osgxMay 11, 2016 at 1:58
I'll need to find a reference for the below (it could be urban legend).
When negotiating the cooperation between Russia and the USA a lot of effort went into defining how everything would be handled. They agreed where training would happen, who would teach, what language they would teach in etc. The language of agreement for training was Russian and at the time the USA spent a reasonable sum of money employing translators to make sure nothing in the training on the Russian side was missed. However, the language in orbit was agreed to be English. That being said a lot of the language spoken between the crew is a combination of the two languages, it's really a 'whatever works' situation.
5$\begingroup$ An aside, perhaps also an urban legend: A technical issue arose at one point in the negotiations over some vehicle (cough ATV). The Russian lead and American lead were both managers (i.e., they were both technically inept). They agreed that a separate technical meeting was needed, involving "three people from Russia, three people from America, and, of course, two translators". The two translators present took umbrage to this: "What? You've always treated us like excrement [another word was used], but this is worse. Now we are not even people!" The two translators then left the meeting. $\endgroup$ Apr 4, 2015 at 14:29