As an appraiser I get this question all the time. Despite the fact that you paid for the appraisal the lender is technically the client and has the rights to the report not you. As an appraiser I can't give you a copy without the lender's approval as it is their report. There are two issues here. A copy and an assignment of the report. Generally, you can obtain a copy of the report by simply asking the lender for it. Some may charge a small copy fee. Assignment is a different issue. If you changed lenders and wanted to take the report to another bank you would have to have the first bank agree to release or assign the report to the new lenders. Most banks won't do this unless they couldn't do the original loan. It is somewhat confusing I know. Call or email me if you need additional information.
When applying for the mortgage loan, it is always a good idea to inform the loan originator that you expect to receive a copy of the appraisal, and have them confirm it in writing. After all, you are paying for it. In most cases, the lender should agree., and see to it that you get a copy.