Yes, it's all in the Privilege Set, Records settings. First, each of the students would need their own Account, a unique string which they would enter to log in. This value would also need to be in a field on their record. Otherwise there is nothing to compare to. Let's call it "Account".
Students would all share the same Privilege Set. In its Records settings, you would enter a calculation in both the View options and the Edit options, as Custom.
Get (AccountName) = Account
In the Fields column you would select, make at least the "Account" field "view only". Be careful using the "no access" setting on fields, as it really means it.
Restricting the View access makes the database much slower. But that would only apply to them, as whatever Privilege Set you used would not have it.
You should also create a script that ran on Startup to take them to their record, an exact Find works; "==" & Get (AccountName). Otherwise they'd see <<no access>> in all the fields; not pretty. After that FileMaker will automatically filter any Finds they do to their own records.