The OdeLog class logs info messages whenever functions are run. However the class documentation states that it is only available in hosted mode. I'm not sure what that means.
It would be useful If I had access to the logs as I interact with the application. Is there any way of doing so? This would probably be possible only if I run the dev server on my own host machine?
Actually, that Javadoc is out of date. OdeLog is presented as part of the debug panel, which is only available when you are using an administrator account in App Inventor. In development mode, you can do this by checking the "Sign in as administrator" box when presented with the dev server's mock Google login page. In production hosting, any user you specify as an owner/admin of the Google App Engine project should be able to access it.
I'd recommend using Ode.CLog() instead. I consider the OdeLog mechanism to be technical debt and would like to see it removed in a future update.