The only time you can be seen as public is if you have a Shared Favorite Trip. If you do not want to be matched with other users, you can either not create a Carpool/Favorite Trip in the system or set it as Shared With "No One". This will then make sure that you are not in any trip search matches.
Even without a Shared/Favorited trip, you can then still log trips and work towards any program in the system you have available.
More on this and other general system information can be found here: