You can use the app without any identifications of yourself (you can use app without login via any accounts).
There are 2 places where other App in the Air users can see very general statistics of your profile: Leaderboard and Nearby widget.
If you do login with Google / Facebook / iCloud, your flight history would be visible in the Leaderboard in the Profile section. However, this is only aggregated data without any details: flight map, number of kilometers (miles), hours, flights, countries, aircraft, airlines, airports. No detailed info would be shown.
Your general stats for the current year would be shown in the Global leaderboard (only First Name, rank, total km (mi) and hours for the current year). You can turn it off.
Go to the Profile -> Leaderboard -> Tap on "..." at the top right-hand corner -> Information -> turn off. Now your stats would be visible only for friends.
Go to the Profile -> Leaderboard -> Tap on (i) at the top right-hand corner -> turn off. Now your stats would be visible only for friends.
"Nearby" is the widget at the Flight Feed which allows you to chat with fellow travelers who are on the same flight as you are or at the same airport (departure / arrival). By default, this feature is disabled (no one can see that you are traveling at the particular flight. You can turn it on at the Nearby widget if you would like to chat with other travelers.