|Unless you are familiar with VBA, I think the simplest way would be to use a VLOOKUP function as I mentioned earlier.|
Since it appears you need to be concerned with both first names and last names, I suggested you add a column where you concatenate these values on both sheets so that the VLOOKUP has a common value to look up.
For example, on the employee sheet (Sheet1), assuming your names start in row 1, you could put this in G1 and drag it down:
=A1&B1 to get SmithJoe
Do the same thing in Column C of the Student sheet (Sheet2).
Then in Sheet1!H1 enter this and drag it down:
=IF(ISERROR(VLOOKUP(G1,Sheet2!$C$1:$C$10,1,0)),"Not A Student", "A Student")
In Sheet2!D2, enter this and drag it down:
=IF(ISERROR(VLOOKUP(C1,Sheet1!$G$1:$G$10,1,0)),"Not An Employee", "An Employee")
Obviously you'll need to adjust your ranges to match the lengths of your list.
Basically what will happen is that if the VLOOKUP function returns an error because it didn't find the concatenated value from one sheet in the list of concatenated values in the other sheet, then the IF statement will be TRUE and return the "Not A..." portion.
If it finds the value, then there will be no error and the IF statement will be FALSE, returning the "A Student" or "An Employee" answer.