Does being sick make you sober?

Technically no, it just makes you stop getting more drunk but it always makes me feel more sober as well. Perhaps it just helps you be more alert.

It gets rid of any alcohol you haven't digested yet at least.

No. Like Brennan said, it rids your stomach of excess alcohol, but there's still too much in your blood stream. You may feel relief at getting rid of it, but you're still very much drunk.