Ft Benning had stalls and the showers were private with a curtain. Our first couple showers were timed by a drill sergeant and you were lined up with soap and towel in hand!
I went through in 2003, also some of my buddies told me Jackson and Leonard Wood were the same. Of course there are different training companies and I cannot vouch for all of them.
My brother went through MCRD San Diego and I am pretty sure he said the toilets were in stalls too.
Probably the biggest problem is shaving in the morning, the sinks were jam packed.
EDIT: So you know the toilets at the range are open, but there are only 8 or so at the largest ranges. The big worry is not about the openness but dropping something into the "well" that isn't supposed to be stored down there. ;-)