First, you should be aware that geo-logging is enabled for the Azure STORAGE service, and not for any other Azure services. Then you should also know that this geo-replication is intended primarily for disaster recovery (and only for today).
If you need to replicate a database (Windows Azure SQL Database, aka WASD), you can use SQL Data Sync - the only known to date synchronize Azure databases (either between different geographic regions, or between Azure and local).
Support for Windows Azure Cloud Service geo-information is not supported. If you need to geographically distribute your application, you need to independently manage the deployment of cloud services in different data centers. If so, for Azure storage I would suggest using one storage service for WRITE operations, but Azure CDN for READ operations. Otherwise, it can become too complicated. Of course, the chosen architectural approach will depend on the requirements of the application (and the expected load).
Then you need to combine the various deployment options with Azure Traffic Manager with tuning the performance algorithm.
EDIT (NOV 2014)
As of Q3 2014, the Azure SQL Database also supports Geo Replication . And Azure Data Sync is deprived and removed service. Azure Storage replication is still offered with three different options: zone redundancy , Geo redundancy, Geo redundancy with Read Access . And there is still no way to replicate geographic regions (i.e., From EU to USA). Replication is still only an option between Geo Zone pairs (the same geography) .
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