The answer is not an easy one.
The only way now to really know is if you have a manual link penalty and you use the disavow file method to remove links and use no other methods for link removal. Then you submit a reconsideration request and Google responds that the request was satisfied - you can be safe to assume Google processed the disavow file successfully.
Another way is if you see your rankings shift significantly a week or so after the file is uploaded.
Right now, when you upload the file, Google doesn't tell you if they processed it. It would be nice to see a status of your disavow file from Google in Google Webmaster Tools but that feature is something not available yet.
Maybe in 2014?
Forum discussion at WebmasterWorld.