Stored locally
That point is at best confusing. Yes, the messages are stored locally. A bit of a security issue, in my mind, since they then can be accessed by other people that have access to your computer. Yes, they are also stored on the server, if you are on an Exchange system. If you are using Outlook as your email client for your hotmail or gmail account then they probably, unless you changed settings, are downloaded to the local machine and then deleted from the server. Also, if people archive (.pst) to the local machine then the message disappears from the server and is no longer accessible online. This last one is the one that confuses more of my users than any other.