Let me choose date of birth as the noun because each person exactly has only one.
Which is the correct sentence? A or B?
A: They write their date of birth on the given form.
B: They write their dates of birth on the given form.
Note: Here I assume that each people gets a separate piece of form so a confusion whether I have to write as follows in case there are more than one forms are given to each of them can be avoided.
A': They write their date of birth on the given forms.
B': They write their dates of birth on the given forms.
Best Answer
Their X just means X belongs to multiple people.
If each of the "their" has an X, then you are talking about multiple Xs, and should use the plural form of X.
If everyone in the "their" group is sharing a single X, you would use a singular X.
An each can emphasize the "everyone has their own X" and can override this.
The second sentence won't raise any eyebrows even though technically it means something unusual.
This could raise eyebrows though: