Carmela seems to be entrusted with important information about the Illuminati. For example, she is aware that Lucius DeBeers is being kept in a secret room behind Everett's bathroom mirror. She insists that JC Denton does not disturb him and, when asked about information on DeBeers, states that she is not permitted to talk about him. This illustrates her loyalty to Everett and to the Illuminati as she does not easily divulge their secrets.
Everett's trust in Carmela is made most apparent by the fact that she is one of very few people that he allows into his home. If he were to allow the wrong person access, he would risk his location being revealed to Majestic 12.