My partner and I get royalties but Getty images only sends a 1099 to one of us. How do we both claim half of the royalty income?

If you received 1099-misc, you will report all this income on schedule C. You will claim the payment made to your partner as an expense. Your partner will report the income that you have claimed as expense on your return.
Read: Your U.S. Tax Return: Tax Filing by Self Employed Sole Proprietor or Independent Contractor