1. View the Edit version of the Google Form, either from your Google Drive, or by clicking the pencil icon of the form's public view:

2. Right click on the 3 dots in the top right of the window. 

3. Select 'Add Collaborators':

4. Enter the Email Address(es) that need Edit Access, and click 'Send':

5. Click 'Yes' if this appears:

6. Click 'Done':

The Google Form is now shared for Edit Access.