I am the co-signer on two student loans but the 1098-E for the loan interest is in the Primary borrower's name. Can I (who actually made the payments) claim the interest?

If you co-signed and made the payments you can enter the student loan interest paid on your own return even if the 1098E only has the student's name on it.  Co-signing means you  are legally obligated to pay.

STUDENT LOAN INTEREST

Only the person whose name is on the student loan and who is legally obligated to pay the loan can deduct the student loan interest.  If you did not sign or co-sign for the loan you cannot deduct the interest.

You cannot deduct student loan interest if you are being claimed as someone else’s dependent, or if you are filing as married filing separately.

The student loan interest deduction can reduce your taxable income by up to $2500

There is a phaseout for the Student loan interest deduction, which means the amount you can deduct gets reduced when your modified adjusted gross income hits certain income levels and is even eliminated at certain income levels -  

• If your filing status is single, head of household, or qualifying widow(er), then the phaseout begins at $65,000 until $80,000, after which the deduction is eliminated entirely.

• If your filing status is married filing joint, then the phaseout beings at  $130,000 until $160,000, after which the deduction is eliminated entirely.

Enter the interest you paid for your student loan by going to Federal>Deductions and Credits>Education>Student Loan Interest Paid in 2019 (Form 1098E)

Look on your 2019 Schedule 1 line 20 to see your student loan interest deduction