I would say yes, but not in front of just any slave, these are their house slaves. The best of the best. The most professional and trustworthy and talented you can find. These slaves were paid well and lorded their status over those lower than they. The best analogy would be to Dowton Abbey or any show about the relation between a household and its servants.

I will add that the city of Rome itself was an awful rumor mill. Just think of the description of Rumor (Fama) in Aeneid 4: Rumor goes throughout the land, a giant winged monster with an eye and ear under every feather, perching itself atop the tallest towers to listen to and spread abroad everything that happens. We can see the Roman rumor mill at work in letters where people ask for news and thus spread it. Cicero has information about many private events like Catiline's meetings or the feast of the Great Goddess, open only to women, that was interrupted by Clodius. In such an atmosphere, hiding things is almost futile. You can (and scholars have) compare it to a small town or rural Mediterranean enclaves where everybody knows everyone else's business. A Roman would tell someone not to bother hiding because the slaves would know anyways.

Romans (and Greeks) set a high moral standard for themselves and thus tried to maintain their reputation. This makes them seem prudish and they certainly guarded their image by acting that way in public. They also feared their house slaves and wives for the power they had to ruin their reputation either through unfounded rumors or by telling embarrassing truths. They thus treated well and acted as virtuous as they could stand. If something did get out, then thy could try ascribing it to the rumor mill. For example, Caesar had to defend against the rumor that he was cuckolded by Clodius during that Great Goiddess scandal. He did so by claiming that it was all ridiculous rumor. He divorced her anyways to save face, or perhaps because she had had sex with Clodius after all. Pompey was often accused of loving his wives too much so says Plutarch. He couldn't stand being away from them and enjoyed their company to the point that it emasculated him. I'm not even going to go into the rumors regarding the emperors from the rich and tabloid-esque biographic tradition.

Who knows where these rumors actually originated. Its a good chance that the servants were part of the process, either originating them or corroborating them. However, I would argue that the rumor mill was relatively harmless to a person's influence and thus they could feel comfortable living openly in front of their servants within social norms. It is only when they got weird (like being cuckolded by a cross-dresser) that things became serious.