The Outlook sync is not automatic. And export to Outlook is a different step than import from Outlook. The developer of Achieve Planner is working on a seamless sync now.
Data is stored locally.
I stopped using the GTD Outlook Add-In once I went to Achieve Planner. Achieve Planner allows many outline levels and the Add-In only allows three levels. So they do not work well together. I have a flat list in Outlook.