Ditto the good experience with OSH Park, they do very good work, very nice quality boards. They do save the boards you submit, they can be seen once you're logged in, so they should only be available to you, but like anything, one should assume a non-zero risk in such a situation. To be clear, that is a general statement and not at all a comment on OSH Park in particular; I have no reason to think they don't do an excellent job from a security standpoint.
I'd also be interested to know where their boards are made. I might ask, they seem like friendly enough folks.