Does the appraisal belong to the bank or the consumer?

For mortgage transactions, the lender requests the appraisal, either directly or through a third party. Even though it's the buyer that eventually pays for the report, the lender is the intended user. The buyer is entitled to a copy of the appraisal - it's usually included with all the other closing documents - but is not entitled to use the report for any other purpose without permission from the lender.

It's different when it's the homeowner hiring the appraiser for things outside securing a mortgage. In these cases, the appraiser may define the purpose of the appraisal; for PMI removal, or estate planning or tax challenges, for example. If not stated otherwise, the home owner can use the appraisal for any purpose.