In order for other staff to invite Students to your Course, they will need to meet 1 of the following 2 sets of criteria:
1. Be an Instance Administrator (Instance Administrators have full visibility and permissions across ALL Courses within Lt).
2. Be invited to your Course as a Staff role that has 'Invite Student' privileges. To do this, you'll need to first ensure that you have created a custom role with 'Invite Student' privileges. To check this / create a role with these privileges, go to the 'Instance Administration' Menu (visible only to Instance Administrators), and go to 'Other Roles'. Within the resulting dialogue, you will be able to view custom roles (and their permissions) and create new custom roles.
Once you've created / found a custom role with these privileges, go to your Course and go to Staff Accounts > Invite. Enter your colleague's details and select the appropriate role with 'Invite Student' permissions. Click 'Send' to invite them.