I ask because I was researching the myth of Persephone and Demeter, and according to a channel called Sarcastic Mythology on youtube, epithets were often used to describe these two and other Greek gods.
This never seemed to be a problem in the religion I was raised in (we follow Vedic scripture). In fact, we actively chant deities' names in order to win their favor/grace.
I would appreciate sources, because when I generally follow a source back to its origin I usually can't find from where the author derived their information.
You could freely use the name of gods and goddesses in Ancient Greece and Rome, as long as you weren't being impious. Some people were even named after the god, Dionysius still being a fairly common Greek name.
Epithets in Ancient Greek had a few different functions. The primary religious function was used to distinguish the different aspects of the deities. Athena was a singular goddess, but she fulfilled many roles such as protector, warrior and craftsman. In order to distinguish these roles you prayed to a specific aspect of the divine. This use of epithets were also used when the Greeks or Romans syncretised their gods with those of another cultures e.g. the British deity Sulis syncretised with Minerva to become Sulis Minerva; or when Alexander (the Great) conquered Egypt he syncretised Ammon and Zeus in Zeus Ammon.
The other function of epithets is purely artistic and descriptive. Greek epics like the Illiad were oral poems before the development of alphabetic writing, and the poets singing these story needed mechanisms to keep time with their meter. So they created these epithet phases that they knew worked, like 'Rosy-fingered Dawn', in order to remember their place in the story and keep time with the poem's meter.
Edit: When I can I will try and provide sources for this, although some copies of the Homeric Epics explain their epithets in the introduction.