I'd say round side, but flat side might be easier. Try one of each and see which is easier.
Do yourself a favor and take a nail punch, or, a 16 penny nail might work, and knock a dimple in the middle of each rivet so that the drill has somewhere to go. That will help to keep the drill bit from sliding off of the rivet as you are drilling.
I understand that it is intimidating, but really it isn't that bad. You'll see. Just don't get in a big hurry and try to force anything. One rivet at a time and don't be afraid to adjust your procedure.
As I grow older, I pay less attention to what men say. I just watch what they do.
- Andrew Carnegie