Friends can see the following in your Peridot profile:
Current Dot information (Age, Name, Archetype, Traits, Distance Walked, and total Dot XP)
When your friend visits a habitat, they may also be able to see any Peridots that you have released there.
Note: Your profile in Peridot is game specific and contains different information than your profile in Niantic Social. Friends can also access your Niantic Profile, which shows recent achievements in other Niantic apps.