When a remote user in VRChat client navigates to my profile and presses View Avatar Details while I am wearing a private avatar, they cannot see the Primary Style and Secondary Style among the tags on that avatar; only custom tags will be visible. However, I can see them locally in "My Current Avatar", "My Avatars" avatar details, and on the website. This also happens in reverse, where I cannot see my friend's private avatar's "Anime" "Human" styles as tags.
This seems to only affect private avatars. The Primary Style and Secondary Style tags of public avatars can be viewed.
A sample avatar IDs uploaded by me, with Primary Style and Secondary Style set (private):
avtr_8a33bffc-c9dd-45a5-a18e-55a18a31c0ef
.
The screenshots attached show what a remote user sees about my avatar and what I see about my avatar in the avatar details.