They can and do if they are reassigned back to headquarters on a domestic assignment. Thousands of people work there and unless a specific individual is being tracked by a hostile power inside the US, it would be difficult to sort out who does what.
There are many levels and kinds of cover. Many CIA personnel are declared to host governments when they are assigned abroad and attached to the Embassy. It facilitates host country protection and working with intelligence officials abroad.
Pfame's usefullness to the Agency as a covert operative was severely damaged when her husband declared in his Op-Ed piece that he was on a mission for the CIA to Niger. His association with the CIA and his new high public profile would make it difficult for his wife to be assigned overseas with her family. She was now Mrs. Wilson, wife of CIA operative, Joseph Wilson.
Pfame's usefullness to the Agency as a covert operative was severely damaged when her husband declared in his Op-Ed piece that he was on a mission for the CIA to Niger.
The disclosure of HIS association with the CIA compromised her ability to function as a spook.
Many CIA personnel are declared to host governments when they are assigned abroad and attached to the Embassy.
But Wilson and the DEMs are trying to plant the false meme that Plame functioned as undeclared. Funny that "covert" has a range of meaning too, as in how many people or which bureacracies know the "secret."