plain red & yellow flag are the colours of the papal state.
other one is the Flag of the Pontificial Swiss Guard
But in the pattern from 1913-today since a flag for them never existed until 1913.
With the coat of Arms of the Guard Commandant Pfyffer von Altishofen, Karl Leodegar (1800-1834)
on white & blue background for Lucerne (Canton of the Guard Commandant).
and the (deformed) coat of Arms of Pope Pius VII (1800-1823)