The entryway/atrium/tablinum section of the house was what clients saw when they came to see their patron for the morning salutatio. It was not wholly public in the way the forum was, but basically anyone could enter the vestibulum and atrium during certain times of day. While most needed an invitation to the tablinum, where the paterfamilias sat, it was still a place where relatively public business was conducted and, as such, needed to show off the status of its master.
Deeper into the house was another public area: the peristyle garden. Its architecture, specifically the colonnades, consciously evoked the appearance of government buildings. Here the family might entertain a large group of important visitors. Like the atrium etc., it was meant to impress upon the viewer the high status and wealth of the house’s owner. Smaller, far more private gatherings might be held in the triclinium, the dining room, used by the master of the house and his guests. Unlike the other rooms discussed so far, this room was open only to those who had a specific invitation.
The small rooms opening onto the more public spaces of the house were used for a variety of purposes, and while they all tended to fall on the more private end of the scale, they varied in status. Some might be bedrooms/boudoirs* or somewhere for the paterfamilias to hold private meetings, while others were used merely for storage. Some houses were big enough to have an entire service complex (kitchen, food storage, laundry etc.). These tended to be hidden from view by long corridors and sharp corners. If the house wasn’t big enough for this, the kitchen could usually be found off one of the more public spaces of the house.** Even the private spaces of the house were private only to the household at large and domestic activities we might consider private, like weaving or cooking, were not infrequently carried out in the more public spaces.
*Only the highest status people in the house might have a bedroom and sometimes even they didn’t. A bedroom, after all, was a waste of space. Better to have a room for another purpose that also had a bed in it. Lower status members of the household would have slept in the more public spaces of the house.
**Precisely which public space varied by house.
Vitruvius, De Architectura (Book VI) - Project Gutenberg
Allison, Penelope M. "Domestic Spaces and Activities." In The World of Pompeii, edited by John J. Dobbins and Peder W. Foss. New York: Routledge, 2007.
Hales, Shelley. The Roman House and Social Identity. Cambridge: Cambridge University Press, 2003.
Wallace-Hadrill, Andrew. Houses and Society in Pompeii and Herculaneum. Princeton, NJ: Princeton University Press, 1994.