If you are logged in that browser and the authentication cookies are present, you should be able to access the services. From the reported browsers, do you have problem browsing the backend or using the services from the backend, since the protected services are used through it. You can try clearing the browser cache and cookies, login again and try access the services. Note that some of the services might not have a default get method and accessing them by url directly will throw an exception of an invalid method.
Do you want to have your say in the Sitefinity development roadmap? Do you want to know when a feature you requested is added or when a bug fixed? Explore the Telerik Sitefinity CMS Ideas&Feedback Portal
and vote to affect the priority of the items