Generally, sympathy and condolences all around. From Gorbachev to Reagan:
We partake of your grief at the tragic death of the crew of the space shuttle Challenger. We express our condolences to the people of the United States and to the families of those who were killed in the accident.
From a joint message of the president of the Soviet Academy of Sciences and the chairman of the Intercosmos Council:
The names of the victims will be inscribed in the history of space exploration.
From Pravda:
In this diabolical whirlwind perished seven people of earth. They perished at the very border of the cosmos without even having crossed it. The television cameras fixed this tragedy for millions of residents of the planet, and for each of us that flaming column, carrying away the lives of those seven daring persons, leaves a deep pain in the soul.
There was some attempts to take shots at Reagan's "Star Wars" defense program, including an editorial that claimed the accident was likely due to the Pentagon causing a "climate of haste and nervousness at NASA, compromising safety", but generally this was the common tone throughout. Keep in mind
a.) The Soviets had the memory of their own disasters, and the US reactions. When Komarov died on Soyuz 1, 47 American astronauts sent a telegram of condolences and James Webb (who ran NASA at the time) called for greater space cooperation. When the Soyuz 11 disaster happened, killing three cosmonauts, the astronaut Tom Stafford flew to Moscow for the funeral as Nixon's envoy.
b.) The year 1986 had thawing relations in general; the two space programs had already collaborated (Apollo-Soyuz in 1975) and Reagan signed a new space cooperation agreement in 1984, Public Law 98-562.
The military dimension to the space programs has always muddled the waters, but in a real sense, all the astronauts were heroes generally, not to just their own countries. Yuri Gagarin is a good example: he went on tour after being the first man in space, and met with world leaders as if they were their equal. He met King Muhammad Zahir Shah in Afghanistan; he met Nehru and Indira Gandhi in India; in England, he met both with Prime Minister Macmillan and Queen Elizabeth; the Daily Mail had the headline "Make Him Sir Yuri!" He was given the Order of the African Star in Liberia.
When the infamous accident of the Apollo 13 mission happened, the desire to see the astronauts return safe was a worldwide phenomenon, with offers of aid from many countries including Russia. Pope Paul VI prayed at the Vatican; there were prayers at the Wailing Wall in Jerusalem. Walter Cronkite said
Perhaps never in human history has the entire world been so united by such a global drama.
Hence, the outpouring of love and support upon the Challenger catastrophe was not a surprise.
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Doolan, K., Burgess, C., Vis, B. (2003). Fallen Astronauts: Heroes who Died Reaching for the Moon. United Kingdom: University of Nebraska Press.
Fraser, E. (2017) "Yuri Gagarin and Celebrity Masculinity in Soviet Culture" from Gender, Sexuality, and the Cold War: A Global Perspective. United States: Vanderbilt University Press.
Kynaston, D. (2014). Modernity Britain: 1957-1962. United States: Bloomsbury Publishing.
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Postnote: The Soviets who were mapping Venus named two of the craters after the two women who died (McAuliffe and Resnik). This naming was mentioned by a NASA book at the same time but the names are not on the IAU official list of Venus craters, so I don't think they were officially approved; all the Challenger astronauts have craters named after them but they're on the Moon. I think perhaps this is because Venus craters are only allowed female names and they wanted to make sure all of them were honored; however, that's mere speculation since I haven't found any discussion of what happened.