We've had that, rarely, where a person didn't want to give out their info and if you don't trust them, I understand but, we have people come to our home to adopt, so trust must be mutual, for starters.
And we aren't in the habit of saving dogs just to hand them to someone just because they
say they are good owners.
We've had people tell us they wont give us info and I'm like, okay, that's fine but we can't give you a dog
If you're that private, then adopting from a rescue may not be for you, but resign yourself to adopting from a shelter, or a sketchy rescue who'd give their dogs to anyone, which means they aren't selective about the type dogs they take in, either (as a rule).
Or, of course, off craigslist.
Just from a rescue perspective