Assuming both of the 1098's are for the same loan/property, the box 3, Mortgage Origination Dates, should be the same on both forms. On the 1098 for the entity that purchased the loan, there should be a date in box 11 which would be the date they acquired the loan. You should enter the information in TurboTax following these rules.
Also, be sure the date is entered in the correct format (mm/dd/yyyy)
Here are the form instructions for more information.