For local testing, you can use a local SQL Server (full, express or even CE) within your app - just use local connection string's in place of the SQL Azure ones in your .cscfg or .config files - then the local compute instance will connect just like any other local process would.
For the most part local SQL and SQL Azure are compatible and interchangeable. Microsoft claim that SQL Azure shares much of its codebase with SQL Server 2008 R2. As a result you can use SQL Server 2008 R2 or SQL Server 2008 Express locally in order to test your application. Indeed, for "simple" applications you should also be able to use the new file-based SQL Server CE.
For "advanced" applications, then there are some limitations with SQL Azure - see limitations on http://msdn.microsoft.com/en-us/library/ee336245.aspx - but for many ASP.Net applications, you should be able to just migrate from SQL express to SQL Azure.
For more details on SQL vs SQL Azure, see http://social.technet.microsoft.com/wiki/contents/articles/comparing-sql-server-with-sql-azure.aspx
For migrating data from local SQL to SQL Azure, see the SQL Migration Wizard on Codeplex - http://sqlazuremw.codeplex.com/