How to securely access the data from another Drupal instance?
The easiest way to do this would be to set up a new MySQL user on Server B that had something like: GRANT SELECT ON serverBdata TO 'serverAuser'@'serverA-IP';

Then you can write a module on server A that connects and retrieves that data.

If that's not going to work, you can look at setting up a menu hook on Server B that delivers the data in a sensible way, such as JSON, over HTTPS. 

if you wanted to get really fancy, you could also encrypt the JSON.