You are trying to access the search result that has is generated by Admin.
Here - admin is the user who owns the search job
- The first "search" is the App name which has access (can be accessible if the job is global or reside within search app)
So if the search job is owned by an admin user you cannot access it with other users.
If the job is accessible by other users then change the username in the URL with a hyphen(-) and give it a try.
In general REST API is accessible to all users, it all comes to down to what objects that user has permission to.