There's already gay men and women in the military. They just aren't allowed to ANNOUNCE that they are gay. (But a lot of people figure it out anyway.) This is called the Don't Ask Don't Tell policy. A person can't be kicked out of the military for being gay, but they can be kicked out if they announce that they are gay, or they are "caught in the act."
I served six years in the Army, as an Arabic linguist. I knew gay people at language school. The military's kicked out several of them. You know how long it takes to train an Arabic linguist? Twenty four MONTHS. And you have to score high on the ASVAB, take an extra test called the DLAB, AND pass an extensive background check. You can't replace military linguists in a hurry. There are never enough of them to go around. Need I point out to you that having folks around who understand the local language in a combat zone might be a little more important to saving the lives of American soldiers, than some privacy concerns?
After my two years of training, I served in the 10th Mountain Division. I knew of five or six gay men and women personally. Nobody in my command cared. Gays and straights alike used to hang out at the local gay bar. They were very supportive of the troops! ;) My command was more concerned about having enough people for our constant combat rotations to worry about sexual orientation.
I think the military's Don't Ask Don't Tell policy is becoming increasingly irrelevant and that by the time they repeal it, it will be NO BIG DEAL to the men and women currently in uniform. Only civilians will obsess about it. The military will accept the change and get on with the mission at hand.
There is no privacy in Basic Training. But that only lasts a few months. Then there is usually some minimal privacy in place. The Sexual Harassment regulations already in place will protect everybody from "unwanted advances" (though God knows straight men are the worst offenders of that policy and always will be.)
Six years in the Army
Two tours in Iraq
We had gay mechanics, medics, linguists, and generator repairers. Nobody gave a hoot.