How exactly are you backing your site up and setting it up locally? The built in import/export tools simply import and export the content. You'll have to setup the user accounts and passwords separately.
If you want to duplicate a site to work on it locally, I would recommend using other plugins made specifically for that purpose which duplicate or backup the entire site (content, themes, widgets, database, etc.) and let you restore it to a different server. Domain name changes are handled automatically.
A popular free tool is Duplicator (http://wordpress.org/support/plugin/duplicator). Very easy to use. If you are duplicating the site locally you'll need a local host environment which you can easily setup with Xampp, MAMP, WAMP, Bitnami or InstantWP.
BackupBuddy is a popular premium plugin that does the same thing plus scheduled backups to any other location (http://ithemes.com/purchase/backupbuddy/). It starts at $75/yr.
If you plan on doing this often, an excellent tool for local development then publishing online is DesktopServer by ServerPress.com. It costs $50 for the premium version and supports importing backups from Duplicator, ManageWP and InfiniteWP. It creates its own hosting environment so there is no need for MAMP, Xampp, etc. There is a Windows and Mac version. The free versions of Managewp.com and infinitewp.com let you do manual backups of your site.