Or perhaps the question would be better phrased: When did having sex publicly stop being de rigueur?
Whoo, a history of privacy question! These are so rare.
Let me say I'll be answering this for Europe, mainly Western Europe (though the trend was pretty consistent across europe), as I don't know much about Asia or African privacy or sexuality. Second, I am answering for post-Roman times, as I'm actually not sure about roman sexuality. I'd recommend you ask /u/heyheymse for more on Roman sexuality.
The short answer is that between the the sixteenth and seventeenth centuries (1500-1600) however, European culture underwent a major cultural shift that would 'invent' privacy.
The reasons for the 'invention' of privacy are many, but let me offer you a few:
The Council of Trent in 1563 demanded that all marriages be in a public place, with at least two witnesses. This had the effect of denying the validity of and private or pastoral wedding that did not have Church involvement. It also increased the emphasis on a 'private' and 'public' spheres.
The 1500-1600 period (and a bit earlier) also saw the seperation of the natural world into 'human' and 'animal' spheres, and animals were increasingly moved out of the home into the pasture or structures built specifically for them. Before this, it was very common for animals to be kept in the home at night.
The architecture of homes changes, as did the purpose of rooms. This period saw better ways to heat the home with central fireplaces and better materials, and saw the 'creation of the upstairs' as Bill Bryson put it--one of my favorite terms. Furthermore, the purpose of the rooms changed:
Bedchambers—and the beds themselves—slowly shifted from being common living areas (in lower-class homes) or sites for social gatherings (in upper-class ones) to being what they are today—private space for the single person or couple who sleep in them. (Ian Moulton, p. 14, Before Pornography)
When life became divided between human/animal and public/private, a similar revolution was happening in the architecture of the house to create public/private spaces--instead of homes being essentially one main room, they became divided up into rooms, and parents began to separate themselves from children, and adults began to have sex in private.
There are, of course, some other details, but please feel free to ask if you have any more questions.