They both didn't do their best.

他们两个中有人没有尽力。both否定的时候是部分否定，如果全部否定应该用：None of them do their best.

这句话不对！要么：Neither of them did their best! 要么：none of them did their best.

Both 用于肯定句： Both of them did their best.或者They both did their best.

他们都没有竭尽全力。

do one's best:尽全力
both:特指两者之间，都

他们俩没有竭尽全力
注意:后面是do "their" best表示两者都没有竭尽全力
楼下胡扯八道

他们俩没尽全力