It is generally understood that Time Lords have regular, Gallifreyan names such as Romanadvoratrelundar, Rassillon etc. Others (mainly 'rogue' or 'renegade' Time Lords) apparently take a name for themselves, based on their chosen path, so we have The Doctor, The Master, The Hermit, The Corsair etc.
Given that The Doctor and The Master knew each other as children, at the Time Lord academy, and before they took their respective paths, it makes sense that they know each other's 'real' names. In the novel The Dark Path, the Second Doctor meets an early incarnation of The Master going by the name Koschei and recognises him from the past, but it is suggested that the name has been assumed and may not be his 'original' name.
At this present time, neither The Doctor, nor the Master's real names have been revealed in the TV show. However, the show and extended media have established that Time Lords have 'house' names - for example, Romana's full name is said to be Romanadvoratrelundar, and she is of the House Dvora. The Doctor Who novel Lungbarrow is named after the Doctor's family house, so this could form part of his real name. However, the Timeless Child story arc which began in season 12, contradicts this novel, and so this may no longer be considered canon, depending on how the arc ends.
Spoiler for season 12:
The Timeless Child story arc, which at this time is still not concluded, has suggested that The Doctor was a foundling and might be of non-Gallifreyan origin. However, as this was told to them by The Master, and with the proof redacted from The Great Matrix, there is some doubt about this. Some fan theories currently suggest that the Timeless Child may actually be The Master. This story may continue and possibly conclude in season 13 (due Autumn/Winter 2021), and could possibly add detail to the house, or family name or whoever is finally confirmed to be the child. For the same reason that I gave for 'Lungbarrow' possibly falling from canon, if the TV show reveals a new origin for The Master this may make the aforementioned Dark Path novel similarly redundant.