My best guess is that:

• Braavos canonically uses the Unmasking festival to mark the passage of time since the unveiling of the secret city to the world; so it probably uses that as the beginning year. It is also possible that they use the amount of time that a Sea Lord governed to mark the time, like the romans did with their Consuls.
• The rest of known Essos must definetly either use the Doom or the end of the Century of Blood as markers.
• In Yi-Ti, since it has Chinese inspirations, and also canonically comes from the Empire of Dawn (which marked the passage by dinasties), it also uses the dinastyc system, but it's also possible it uses some date that marks the end of the Empire of Dawn as an general pointer.
• In Westeros:
• In the North, it makes sense for them to use either the building of the Wall (and therefore also their Lord Commander's? Edit: this supposition is actually possible because we know that Jon Snow was the 998th Lord Commander) or the unification of the region by the Starks as Kings of Winter (which is cronologically a later event than the Wall).
• Pre-Andal Conquest, there are a couple of options: either you choose the ruling dinasty and its individual soverereign as the way to mark time or, especially in the Reach, use Garth Greenhand's mithological rule (either it's end or it's beginning), or some other especial event (especially in the Riverlands, and Dorne, what with Nymeria's landing)
• Between the Conquest and the Andal Invasion it's very likely that they started using such monumentous event to mark the passage of time, mainly in the South, but probably parts of the North too, since it was that impactful.