Can a plural possessive pronoun be used for “side”?

Be the following paragraph:

Finally, the storm ended. Then, the two groups of men reached the centre of the field with their goods to trade. After they did it, each side took what was theirs and went back.

Is it correct to use “theirs” in the third sentence?

Furthermore, is there some difference between British and American English in this regard?

I will appreciate your help.

P.S.: Perhaps further clarification of the context is needed. The two groups are made up of men of different nationalities, speaking different languages, and armed.