There is a setting on each membership record that allows user to configure who to "Send renewal notice to" - the giver, the primary member or both. Here are steps to locate and edit this setting:
  1. Go to the constituent record
  2. Go to the Memberships tab
  3. Locate the gift membership and click the Other Tasks button
  4. Select Edit renewal information
  5. Change the "Send renewal notice to" drop-down
  6. Save
This setting will be followed anytime the membership renewal notices process is run and includes a gift membership renewal notice.
View this idea about setting the gift membership default to 'primary member' for back office payments.

If you need to change a large number of membership records, use these steps to build a query and browse through: 
  1. Go to Analysis > Information Library.  Click Add an ad-hoc query.
  2. Select the source view of Memberships and click Ok
  3. In the middle, select Is gift and move this to Include Records Where.  Set the criteria to is equal to Yes
  4. In the middle, click on Send renewal notice to and move this to Include Records Where.  Set the criteria to either is equal to Giver or is equal to Primary Member, whichever you are looking to change.
  5. Go to Set Save Options tab.  Name and save your query.
  6. Run the query and double click on each line of results to easily browse to the membership record.  Then use the steps above to edit the renewal information.