The 172s my club uses were LAME modified (standard door hinges replaced with modified quick release ones) to allow quick removal of the doors but only some of the clubs instructors have official sign off in their logbooks to remove them. Not sure how much of this is similar in Aussie but it wouldn't surprise me if its even more convoluted.
The NZ FM supplement for door removal in 172s only allows the right hand door to be removed.