You need to assign Staff user accounts to a site in order for the site to appear on both the Handover app and the Inspection app for that user.
At the bottom of the Site detail section of the site there is a box displaying all Staff accounts currently assigned. If there is no staff assigned, click on the Green “+ Add Staff” button in the centre. Otherwise, at the top-right is an Add staff button which allows you to add Staff user accounts. Next to each Staff name is a bin icon to remove the staff member from the site as well if required.