I think we can assume that agents attached to embassies are probably well known to the local intelligence agencies in whatever country they're in. This is absolutely fine from the agency's point of view, since the CIA needs a big operation in Beijing (and conversely the Chinese need a big operation in Washington) and it makes much more sense to run it semi-openly within the embassy, since the alternative is to attempt to run a big covert operation outside the embassy.

I think we can also assume that in countries where it matters there are probably spies who are not employed by the embassies and who may or may not be known to the local intelligence agencies.

I guess in conclusion, you shouldn't go around feeling all clever just because you've figured out some secret that you think some intelligence agency didn't want you to know. They have layers upon layers of secrets, and some of them are designed to be discovered.