OSX App Store updates with multiple users
Applications downloaded from the App Store are available to all local users, there is no way (and also no real reason) to buy and download the same application twice. For updates you must be logged into the Store with the Apple ID used for the initial purchase.

In the situation described in the question the asker can


log into the Store with the "other" Apple ID to get the download
ask whoever purchased the application to log into his/her account and get the update