GNOME Evolution downloads content of remote accounts to the local machine for quicker access to it. The usual place is under ~/.cache/evolution/ (more information about where the user data is stored can be found here). Evolution itself doesn't share the local data with any 3rd-party, except as directed by the user and being designed and known to access it through evolution-data-server interfaces.
In case of mail accounts only basic information about the messages in respective folders is stored locally. The message content is downloaded on demand and resides in the local cache for a limited time, unless user has set Evolution to store the messages for offline use, which will download them as soon as possible and will keep them stored locally until the message is deleted from the server either by Evolution itself or any other client. Sent and Draft messages can be stored on the server, as configured by the user. The messages and folders of remote accounts can be fully managed by Evolution.
The other than Mail accounts (meaning Calendars, Contacts, Memo Lists and Task Lists) copy remote content on the local machine too, if possible. Similarly as with mail, the information resides in the local cache until the item is deleted from the server.
New Account Lookup
When creating a new account (Mail or Collection), the settings can be looked up for with the user's email domain part. This is usually enough to receive information to configure both the Receiving and Sending parts. The full email address is never used, rather a the user name is change to a fake value in any such queries. Users can disable this lookup during the new account wizard.
Evolution can be configured to show sender's photograph, which can be looked for on the libravatar.org server. This lookup doesn't expose sender's address, it asks the server with a checksum of it. To disable this lookup open Edit->Preferences->Mail Preferences->Headers tab, and disable the Search libravatar.org for the photograph of sender option. The option doesn't do anything when the whole lookup for sender's photograph is disabled.