It is important to note that Greek and Roman homes were not the same, that both developed chronologically and geographically (more so in the case of Roman houses), and that they varied between social classes.
One common theme, however, is the extent to which men and women were separated. In ancient Greece, a room called the andron was set aside for the men and their male guests. This room often had a separate entrance so that male guests could enter without the possibility of encountering the women of the house. In houses with two floors, the women would usually have a space upstairs with the children.
Roman men and women lived together more freely but still had separate parts of the house dedicated to their use. Unless they lived in cramped quarters, husbands and wives did not normally sleep together. A wealthy couple would often have a series of three rooms, two of which would serve as separate bedrooms to which they could retire.
The concept of privacy as we now understand it would have been alien to both Greeks and Romans. Even in large Roman villas, there is no distinction between private and public spaces, and any room could become a bedroom for guests. Slaves were not expected to have any private lives and sometimes slept outside their master's door.