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.