Can Regional security officer(work at embassies) ask the host government for privat informations that are only allowed for judge's and law enforcement agencies about a candidate for a background check ?
Deciding if the officer can ask the host government is probably a question of administrative procedure in the embassy.
Deciding if the host government can answer and reveal usually secret or confidential information about their citizens is up to the host government and their laws and constitution. Most nations reserve the right to cooperate with foreign intelligence agencies, and to pass information about their own citizens or third-country citizens.
What will happen is another question. The Five Eyes Group, the NATO, and the EU are routinely cooperating on many intelligence matters, but they are also democracies with laws to safeguard citizen's privacy.