The first Soviet missions (Sputnik, Vostok 1, etc.) were generally secret beforehand. The purpose of this was to allow the denial of failures; the USSR only announced successful launches. Soon, once spy satellites were passing over Baikonur, the US could tell that the Soviets were going to launch something, but not what it was (it's rather difficult to hide an R-7).

The Soviets had more warning, due to differences in society. The US announced launches beforehand, and the public even came to view them, bring the kids for a picnic, etc. Something like that would never happen, and is still difficult, at Baikonur.

Now, your question extends to later parts of the 1960s as well. At that point, there was some communication directly between astronauts and cosmonauts, notably some face to face meetings at some of the Paris International Airshows. Later, meetings became more common, and some astronauts were given tours of Star City, the cosmonaut's training facility, while some cosmonauts visited astronauts in the United States, especially in the run-up to the Apollo-Soyuz Test Project. An interesting note was that in 1971, astronaut Thomas Stafford served as a pallbearer at the funeral of the crew of Soyuz 11, who perished when their craft depressurized during the activation of the reentry sequence.

I'm sourcing most of this from a wonderful book, Two Sides of the Moon: Our Story of the Cold War Space Race which was co-written by astronaut David Scott, commander of Apollo 15, and Alexei Leonov, the first person to perform a spacewalk as well as the Soviet commander of the Apollo-Soyuz Test Project.

On spies — the US generally had very poor human intelligence regarding anything in the Soviet Union. They relied primarily on technical intelligence (e.g. decryption, wire-tapping, spy planes, spy satellites) for their information. They later were able to get a couple useful moles, but all of their human networks eventually fell apart, and so did some of their technical means (notably the wire-tapping) because the Soviets had much better human intelligence throughout the Cold War, including agents within the US military and intelligence structure (so the Soviets would be tipped off of American spies or wire-tapping ahead of time, whereas the US was never quite aware of the depth of Soviet spying).

So with regards to space things, and atomic things and other secret military things, the US was relatively "blind." If they couldn't figure it out from looking at things from high above, or listening in, they didn't know it. There are, as noted, some exceptions to this, but as a general trend it holds true.