Follow-up: Is this why some Roman houses had warmed floors (hypocaust) or was this completely unrelated to whether they had shoes on or not?

/u/notavriend asked a question that I think enlightens yours, so while I can't answer decisively what Romans wore on their feet (EDIT: Now I can, see below...) I will reply to his question as well as yours and hopefully give some useful information. Keep in mind my source is "Daily Life in Ancient Rome" by Jerome Carcopino, published in 1941, so it is possible that archaeology has further enlightened some of the stuff I paraphrase from him below between then and now. Perhaps /u/Tiako could elaborate further.

Regarding the warmed floors of very well-equipped Roman houses: The hypocausis (the heating apparatus) and the heat-chambers (hypocaustum) that sat between it and the heated room (suspensurae) did not really qualify of central heating. It was a furnace that sat below one more heated rooms yes, which had a flooring of "bricks, a layer of clay, and a pavement of stone or marble." This floor was designed to slow and moderate the rise of temperature. However the heated surface was not greater than the surface of the hypocaustum, it was not applicable to multi-storied buildings, and it was never used to heat an entire building unless it was "one single and isolated room like [a] latrine". Even in the buildings that did have such a furnace system, "it never occupied more than a small fraction of the house: the bathroom in the best-equipped villas of Pompeii or the caldarium of the public baths."

This furnace system did not seem to exist at all in the insulae in which so many Romans lived. The inhabitants of the insulae also could not gather round a fire as could the rural peasantry, and so the insulae, also not possessing proper windows and a lack of lighting, were particularly vulnerable to extremes of temperature and their inhabitants must have worn something over their feet at times of cold, though I've never seen this specifically discussed. Even proper houses, in protecting themselves from the elements, had to sacrifice natural light. Windows were rare in Rome and were not anything like as good as modern windows (due to size and quality/thickness of glass it seems), with hanging cloths or skins or folding shutters to cover the window-like openings of their houses which "while keeping cold and rain, midsummer heat or winter wind at bay, also excluded every ray of light. In quarters armed with solid shutters of this sort the occupant, were he an ex-consul or as well known as the younger Pliny, was condemned either to freeze in daylight or be sheltered in darkness". The insulae were no better off, lacking even fireplaces. Plants, such as pots of flowers and trees seem to have been common around houses however, and that might have improved the effect somewhat.

It's my understanding that floors in decent houses were stone on the bottom, perhaps with concrete and a mosaic finish. On the upper floors they would have to be wood, and apparently they could also have a concrete layer. Perhaps non-mosaic floors might have a mat or carpet placed over them in some instances, but I don't know anything about this. In the interests of practicality, particularly in the insulae, I find it hard to imagine the inhabitants would not do everything they could to keep themselves warm, whether by wrapping their feet and bodies or putting skins on the floor. I would expect that on stone/concrete/mosaic floors that often get cold a Roman would generally wear sandals, but I confess that this is just speculation from practicality. On the other hand, that some floors were heated in nice houses and public baths might suggest that they did get around barefoot also. So in the end, I don't actually know whether Romans generally wore shoes indoors or not! On a final note, this website claims that the Romans wore sandals indoors. This wouldn't surprise me, but like almost all websites it's short on sources and known authors (Linda Alchin isn't exactly a Roman specialist) so I wouldn't rely on this at all.

EDIT: Finally, a half decent source, and probably Linda Alchin's source for the final paragraph above. LacusCurtius.org, about as reliable as websites get, citing "A Dictionary of Greek and Roman Antiquities" (1875), says that soleae were sandals specifically designed for indoors use. Note the following excerpt:

"The solea, as worn by the upper classes, was adapted chiefly for wearing in the house, so that when a man went out to dinner, he walked in shoes [Calceus], taking with him slippers [Soccus] or soleae, which he put on when he entered the house. Before reclining at table, these were taken away by a servant (see woodcut, p308; Plaut. Truc. II.4.16; Ovid. Ar. Am. II.212; Mart. VIII.59.14); consequently when dinner was over it was necessary to call for them (Plaut. Truc. II.4.12, Most. II.1.37; Hor. Sat. II.8.77). But, according to the state of the roads or of the weather, the shoes or boots were again put on in order to return home, the soleae being carried, as before, under the arm (Hor. Epist. I.13.15)."

All the literary references are right there in the text as you can see. There is more on this type of footwear here. So essentially I wasted my time writing much of that post above! Upper-class Romans wore a specific type of sandals, which could even be called 'slippers', inside the house, and even carried them around with them when moving from dwelling to dwelling, because they had completely different footwear for outside the house.