The Distributive Plural
When each part of a plural subject possesses something individually, the thing possessed must generally be in the plural as well. For example:
The two women blew their noses.
Each woman possesses one nose, so, logically, two women possess two noses. Some usage experts call this type of plural “the distributive plural” (Quirk et al. 768).
The Distributive Singular
But in several cases, the thing possessed should be in the singular.
To Indicate Joint Possession
For example, use the singular when two people possess something jointly:
While traveling together, the two women got lost and consulted their map.
To Convey Universal, Abstract, and Figurative Ideas
Wilson Follett remarks that the thing possessed also “remains in the singular when what is plurally possessed is universal, abstract, or figurative” (211). So if, after blowing their noses, our two women celebrated with a bottle of wine, we might say, The two women toasted their health (universal). If the doorbell rang while they were drinking the wine, we might say, The two women were led by their curiosity to open the door (abstract). But if no one was there when they opened the door, we might say, The visitors wanted to get something off their chest but had a change of heart (figurative).
To Avoid Ambiguity
Quirk et al. observe that sometimes the singular is needed if the plural would be ambiguous (768):
We asked the children to name their favorite animal.
If they were asked to name their favorite animals, the children might not be sure if they should name more than one. To make clear that each child, rather than the group as a whole, should give an answer, we could revise as follows:
We asked each child to answer the question, What is your favorite animal?
Mind Your Nouns
As Words into Type warns, when the sentence has more than one noun, you must be careful to use the singular for the correct noun (357): You should have seen the expression on their faces when they heard the news (not the expression on their face).
Follett, Wilson. Modern American Usage: A Guide. Revised by Erik Wensberg, Hill and Wang, 1998.
Quirk, Randolph, et al. A Comprehensive Grammar of the English Language. Longman, 1985.
Words into Type. 3rd ed., Prentice Hall, 1974.