How can I then access my maps saved to publilc server?
1. You can save the map from one computer to your hard disk. Then send it to new computer via email, via email or so. Then open this file from the new computer.
2. You can save the map one computer to your 'Google Account'. Then later open the map in the second computer via the same Google Account you had saved in previous step.
Hope your doubts are cleared.