How do I know how much of this beneficiary's education expenses were used to claim an education credit on someone else's return?

You'd get this information from the tax return of the person who is claiming the education credit. This is usually the student or the student's parent who is claiming the student as a dependent. Where to find this amount depends on if the American Opportunity Credit or the Lifetime Learning Credit was claimed for the student.

• American Opportunity Credit (Most common education credit claimed)
  Enter the amount from the other return's Form 8863, Part III, Line 27 where the student's name is shown on Part III, Line 20 (maximum $4,000).
  
  Example: The beneficiary of your 529 plan is your grandson. You're the recipient listed on your Form 1099-Q. Your grandson is claimed on his parent's return as a dependent. The $2,500 maximum amount of the American Opportunity Credit is being claimed on his parent's tax return based on your grandson's college expenses. You would enter $4,000 (The amount shown on his parent's Form 8863, Part III, Line 27 where your grandson's name is on Part III, Line 20).


• Lifetime Learning Credit
  Enter the amount from the other return's Form 8863, Part II, Line 11 (maximum $10,000).
  
  Example: The beneficiary of your 529 plan is your grandson. Your grandson is claimed on his parent's return as a dependent. The $2,000 maximum amount of the Lifetime Learning Credit is being claimed on his parent's tax return based on your grandson's college expenses. Your grandson is the only student whose expenses are used to claim an education credit on his parent's tax return. You would enter $10,000 (The amount shown on his parent's Form 8863, Part II, Line 11).
  
  What if there are multiple students claiming the Lifetime Learning Credit on the other return?