Great question. /u/restricteddata could probably give you some more information, but I've been researching this topic lately for a paper. The Soviet Union's principal ICBM range was (it's still in operation by Russia) the Kura or Kama range in Kamchatka. Typically, missiles would be fired from European Russia or central Asia and launched into the range offshore.

The United States had an intense interest in the capabilities of these missiles, and it monitored them closely with radars stationed in Alaska. Submarines and other seagoing vehicles were used to recover missile fragments as well.

The first Soviet ICBM, which began flight tests in May 1957 and had a successful long-range test in August 1957, was announced as ready at the end of August. It wasn't until the October launch of Sputnik that many Western observers took the Soviet announcement seriously.

Missile accuracy is measured in what's called CEP, or Circular Error Probable. If you draw a CEP circle around your target, 50 percent of your shots will be within the circle, and 50 percent will be outside it.

The R-7, the first Soviet ICBM, had a CEP with a radius of between 2.5 kilometers and 5 kilometers.

On the American side, the Atlas was the first successful U.S. ICBM. Its first model had its prototype flight in June 1957, but it did not go well, and the missile was destroyed in flight. Subsequent launches in 1957 and 1958 were more successful, and the missile entered service in 1959.

American missile tests were flown from Cape Canaveral in Florida and Vandenberg Air Force Base in California. There were several impact ranges, including in the Atlantic Ocean, Indian Ocean, and Pacific Ocean.

In the Atlantic, the Ascension Island "splash net" used an array of sonobuoys to pinpoint the landing spot with remarkable precision.