To add to fgysin's answer, if your campaign takes place over longer scales, with travel taking e.g. weeks, quad paper becomes useful:

In this case you'd use either a separate sheet, the other side of the paper or the right half for notes. Cross over days that go past with nothing significant happening, but use a number for notes if something is important:

Depending on the size of the paper and grid, you can fit a year on 2-4 pages, meaning even decades-long campaigns stay (relatively) organized.

Regarding who maintains it, as GM you can more easily set yourself reminders in the future or notes about NPCs if you do it. However, if a player does, you can still ask them to mark down e.g. a star without telling what it means. Sneaky way to hint at a deadline...

To add to fgysin's answer, if your campaign takes place over longer scales, with travel taking e.g. weeks, quad paper becomes useful:

In this case you'd use either a separate sheet, the other side of the paper or the right half for notes. Cross over days that go past with nothing significant happening, but use a number for notes if something is important:

Depending on the size of the paper and grid, you can fit a year on 2-4 pages, meaning even decades-long campaigns stay (relatively) organized.

Regarding who maintains it, as GM you can more easily set yourself reminders in the future or notes about NPCs if you do it. However, if a player does, you can still ask them to mark down e.g. a star without telling what it means. Sneaky way to hint at a deadline...