Behind the scenes, Appointiv uses Contact records to represent Service Providers for bookings even if the Service Provider is a Salesforce User. So, for that reason, if you are making Contacts Private, you may need to create a way to share the Service Provider Contacts with everyone.
To do this, create a new Account - call it Appointiv Service Providers for simplicity.
Find all the Contacts that represent your Service Providers and add them to this Account. (search for the field "type of record" = "service provider" to find them easily)
Create a sharing group. Call it Appointiv Service Providers...or something equally obvious.
Add all your users to the Sharing Group.
That should solve the issue.