Using measure words for "one [thing]"
The full version is 一个人说一句话. 句 is a measure word here. 一人说一句 follows a speaking cadence. It can also be briefed as 一人一句 colloquially when context is clear(we know the intended action is - 说). The purpose of this omission is to make it follow a kind of speaking pattern '一人' 说 '一句'. Another example is 一日一字(a character, a day) which is the title of a TV program in the long past. 

We don't usually omit 个 in a full sentence especially in writing. E.g. 一个人的日子不好过. You would probably only see 一人的日子不好过 in a poet or lyrics. 

It's interesting to know that 一 xxx 一 xxx is a pattern to express an idea briefly. E.g. 一人一个，一人一句，一人一本，一日一练，一天一根，etc.

Hope this could help.