How can I verify if someone has actually gotten the degrees from a university that they claim they went to?

Assuming you are an employer, and not just a curious individual, you can ask them to have their transcripts sent to you. The potential employee orders them from the university, so they pay for them (but the cost is very minimal), and the papers go directly from the university to you, without passing through the hands of the potential employee.

If your interest is personal curiosity, you will have more trouble, because laws (FERPA) keep schools from telling you anything about their students and graduates.

You have to ask the person for a transcript - which depending on the university could cost anything from $5-25 dollars but they send it to the person and you get it from them. Or you can pay the money yourself at their university and get it - either way, but if its for a job you want them to pay.

If you are an employer or a law enforcement officer, the university's transcript dept. will confirm this. If you are not one of those, they will not.

check the school's alumni directory?