1. Students on Earth may also play soccer with students who live on Mars.

2. The students on Earth may also play soccer with the students who live on Mars.

[Are both okay? Do they have the same meaning? Does 'the students on Earth' mean 'all the students on Earth'? Does 'the students who live on Mars' mean 'all the students who live on Mars'?]

They mean the same, and both are acceptable. I question "also", though. Who else are students on Earth allowed to play soccer with, students from Neptune, other Earthlings, or just whom? You can leave out "also" unless you want to explain.