How private are things like showering?

It all depends on where you are based. During basic training and initial trade training all the showers are in one room (shared) but have shower curtains for privacy. On completion of training and on assignment to you first unit, you may have the option of having your own room which means you will have your own bathroom complete with toilet, sink and shower. It all depends on where you are posted and what the availability is like at that unit.

It's very rare now that you would ever, ever shower in one open plan (same gender) shower!!! Long gone are the days that that's appropriate! Most showers now are cubicles - all be it next to each other but you are in your old little (Sometimes!) lockable unit! Military Gyms have separate showers or showers, as Matt said, that are just coned off by a curtain each. Sounds crazy now, but it does get to the point during training that you walk into the loos (as females) and everyone's got the door open to their own loo having a natter to the people cleaning their teeth for example!!