Clandestine call centers are non-official secure message relay and telephone surveillance services operated by the KGB.
The call centers act as a secure middleman between the KGB and it's agents. Mostly seen to be delivering instructions to agents via the use of seemingly innocent sounding phone conversations, they also act as a surveillance hub for the phone conversations of the illegals and their agents.
They are capable of setting up secure lines between any numbers via the use of a custom switchboard and many different handsets. In Echo, George attempts to destroy the switchboard when ambushed by Andrew Larrick.
While there are likely many more, only two clandestine call centers have been shown in The Americans - the first run by George; another is operated by Joan after George's death at the hands of Andrew Larrick. They appear to be the only person manning the phones.