If you have someone's IBAN number in another European country than the one you're in (therefore, no local branch of their bank of where you are), is there a way to find out the full name of the person that the IBAN number belongs to, so as to ensure that it is the correct person to transfer money to?
No, you cannot. That would be a big breach of all kinds of privacy laws.
What you can do is give the name of the account holder when you transfer the money.
Then the bank should reject it when you put in "John Doe" but the account is registered to "Regina Miller".
But they will never give you the data.
(Just for the record, IBAN accounts are not anonymous. If the police comes knocking with a warrant, the will get the name and all data easily. But a private person has no right to get this data from the bank.)