how can I see the arguments passed to a stored proc on Azure?
Today, I think you would have to add logging to the stored procedures (or to the calling application). Trace / profiler are not supported on Azure SQL Database, and never will be.

At some point in the future, you should be able to use database-scoped Extended Event sessions. Bob Beauchemin talked about this here and here, and there is an official marketing statement from Microsoft's Tiffany Wissner here (but nothing technical):


Preview Available for Latest Version of Azure SQL Database


If you have an Azure database and you happen to be on the most recent public version (13.0.15 as far as I can tell), you can start to see some evidence of her statements about Extended Events. For example, I have a context menu item that I can't do anything with yet:



So, it's coming, just not yet. Until then I can't really think of a decent way to obtain runtime parameter values for called procedures unless you change the application to retrieve actual, post-execution plans and log the XML output somewhere. And if you're going to do that, well, you may as well just add the logging of the actual values - much easier to consume.

This answer, of course, is subject to change as it gets older.