I was reading an article on Wikipedia and came across a picture of "the Czech (formerly Czechoslovak) embassy in Berlin". Right next to this section of the article. I then wondered: did they split up the embassies between the countries? If so, how, and if not, what did they do instead?
Federal Czechoslovak buildings were distributed according to a 2:1 ratio (the respective size of Czech vs. Slovak Republics). This extended to embassy buildings as well.
In Paris, Berlin, and Moscow the Czech Republic retained the former Czechoslovak embassies, while the Slovaks built new. In Paris, the Czechs received a beautiful palace, while in Berlin they received the finest 1970s Eastern Euro diplobuilding possible.
In London, the Czechoslovak embassy consisted of two buildings: the actual working offices of the embassy, and a separate and larger residential/commercial office building. The complex was split, with the Slovaks getting the the embassy building, and the Czechs receiving the larger office building.
North America presented an interesting trade-off between the two new countries. The Czechs got the larger embassy in Washington, DC, while the Slovaks obtained the smaller building in Ottawa. Each obtained new offices in the respective other countries.
The foreign assets of Czechoslovakia mostly went to the Czech Republic, which was seen as the successor state. Slovakia, for the purpose of international relations, was a new state.
Businesses and industries not yet privatized were split by location, and new national bodies were set up to run monopolies like the trains and telecoms.
The same process happened in sport. After the breakup, Czech Republic kept Czechoslovakia's slot in the top tier of international ice hockey, for example, while Slovakia, a hotbed of hockey talent, had to start in the fourth division and win their way up.