Early warning satellites (the kind that stare at a region and watch for launches) weren't quite practical yet, so the primary way to detect an incoming missile was by ground-based radar - and that wouldn't necessarily pick up a launch initially. Once something was detected, it could be classified fairly quickly as a space launch or as an attack by its trajectory. So I don't think it was usually a big deal and nowadays I believe there's a notification system in place (especially important when testing actual missiles).

However, once missiles on subs became available things got a little scarier, as the flight times were short. There was a fairly serious incident in 1995 where the Russians mistook a sounding rocket as a submarine-launched missile.