I downloaded the latest stable JOSM package last night, after being prompted to update to the latest version, but then was not able to run the application. I received the rather cryptic error message "JOSM.app is damaged and can't be opened. You should move it to the Trash." What this actually means is that the Gatekeeper function in OS X 10.8 Mountain Lion was blocking the (unsigned) JOSM software.
To get round this, temporarily disable Gatekeeper by opening System Preferences > Security & Privacy, and set 'Allow applications downloaded from:' to 'Anywhere'. Once you open the application, you can re-enable gatekeeper, and it appears that JOSM will open again without a problem. But this will likely have to be done for every update :(
I'm definitely not trying to blame the JOSM developers for not paying Apple for a developer licence to sign the software. In fact, they do a great job in continually developing the software to make it easier to edit OSM. Just thought I'd post this, in case it helps anyone else.