I think you've made a big leap in concluding she thinks you're all liars. She might just be a very private person.
Did she call the children liars? To their faces? To your face? Or is that your interpretation?
I would take the fact that they are nice to you all and have been over a period of years over the fact that she is obviously paranoid about privacy in her personal space.
There is obviously stuff in there that she doesn't want to be seen - I wouldn't start speculating what it is, that way madness lies - but I wouldn't react by ruining the family relationship over it.
Yes if I was 100% sure that my children were telling me the truth then I'd be miffed, but that is enough.