I've tried putting a calibre generated mobi on two Kindles and an iPad. I loaded both Kindles through calibre and copied the exact file from one of the kindles onto the iPad through the kindle app file sharing section in iTunes. I also verified that there was a GUID in the ASIN metadata field using Mobi2MobiGUI.
The Kindles sync properly. The iPad doesn't. I also tried loading the same file through dropbox and it still didn't work. Books purchased through Amazon do sync, so it isn't that the feature is completely missing on the iPad. There must be something else going on.
Has anyone been able to get this working with the Kindle app on an iPad?