Nothing is covered up that will not be revealed, or hidden that will not be known. Therefore whatever you have said in the dark shall be heard in the light, and what you have whispered in private rooms shall be proclaimed on the housetops. (Luke 12:2-3)
Being a person with integrity doesn't just mean you're a good person. No, if a person has integrity it means they are integrated - hence the word. Their mind, body, and spirit are intertwined together into a coherent personhood. They know who they are, and they are that person.
This means a person of integrity isn't one person when playing poker with the boys and another at preschool drop off with their daughter. This means what they have to say to you they would say to your face, not to someone else and not to a computer screen. A person of integrity should have relatively consistent behavior and you should know where you stand with them. They are not twenty different people nor are they unsure as to who they are at all, so while they may not be perfect saints they are surely consistent.