So by default your name and DNA are not visible to anyone.
You'd have to make your profile public in order for other people to see any information about you. All it shows in the relative finder is that there is a match. The person just follows the "contact" link and sends you a message. They don't really expire, so you can just sit on them until you feel like responding/granting access-- at which point they can then see your name.
At that point they can't see your DNA either-- you have to go through another process to grant them access to your DNA.
All in all it's a pretty good system.