For your example, you should definitely use the plural, however there can sometimes be ambiguity in English. For example, how many legs get broken in this sentence?
John and James collided on the ski slope and broke their legs.
A better version would be
John and James collided on the ski slope and each broke his leg.
But what if it's John and Jane?
John and Jane collided on the ski slope and each broke their leg?
or
John and Jane collided on the ski slope and each broke a leg?
Neither works perfectly, in my opinion, so it can be an awkward way of saying things.