Different from changing your own profile picture
This guide is about saving someone else's photo to your device, not updating your own -- changing your own profile photo is a completely separate settings flow.
Privacy settings can hide this option
If the contact has restricted who can see or save their profile photo, the full-size view or save option may not appear at all -- this feature only works when their privacy settings permit viewing their photo.
Frequently Asked Questions
Will the contact be notified if I save their profile photo?
No, viewing and saving a profile photo happens silently on your end -- the other person is not notified.
What if the save option does not appear when I long-press the photo?
This usually means the contact's privacy settings restrict who can view their profile photo at full size, in which case saving it is not possible.