Sorry, I should have mentioned that we're using Axis. I have a handle to the org.apache.axis.client.Call object, and I tried getting a handle on the SoapMessage object by doing _call.getMessageContext().getMessage(), but that returned null. getRequestMessage() and getResponseMessage() returned null too.
I guess the right way to handle this is to find the trigger in the server config to turn on logging of soap calls. But the reason I'm looking for an alternative fix is because no one seems to know where or what that flag is.