Say that someone has a French visa for study purposes. If he wants to travel to Germany by train for instance, can he go there freely without a passport or visa check?
Yes you can, if you have a long term visa from one of the Schengen countries you can freely move to any other Schengen country and stay for a period of three months or less in any 6 months period (90/180 rule).
From EU Immigration Portal:
When you stay in an EU country for a long stay, usually for more than three months, you will generally be issued with a national long-stay visa and/or a residence permit.
If your long-stay visa or permit has been issued by a Schengen area country, you can travel to another Schengen area country for three months per six-month period