Odd and even itemize symbols

You can make a command that changes its own definition, and use that as the label, for example like this:

\documentclass{article}

\usepackage{enumitem}
\usepackage{fontawesome}
\usepackage{xcolor}


\makeatletter
  \def\speech@bubble@a{%
    \color{blue!70!white}\faCommenting
    \global\let\speech@bubble\speech@bubble@b
  }
  \def\speech@bubble@b{%
    \color{green!70!black}\faCommentingO
    \global\let\speech@bubble\speech@bubble@a
  }
  \newlist{dialog}{itemize}{1}
  \newlist{dialog*}{itemize}{1}
  \setlist[dialog]{
    before={\global\let\speech@bubble\speech@bubble@a},
    label={\speech@bubble},
  }
  \setlist[dialog*]{
    before={\global\let\speech@bubble\speech@bubble@b},
    label={\speech@bubble},
  }
\makeatother

\begin{document}

First person first:
\begin{dialog}
  \item bla
  \item bla
  \item bla
  \item bla
  \item bla
\end{dialog}

Second person first:
\begin{dialog*}
  \item bla
  \item bla
  \item bla
  \item bla
  \item bla
\end{dialog*}

\end{document}

Here is a solution with the \newtoggle command from etoolbox. To simplify the use of this solution, I defined a new itemize-like environment, alternitem:

\documentclass{article}
\usepackage{enumitem}
\usepackage{fontawesome}
\usepackage{xcolor}
\usepackage{etoolbox}
%
\newtoggle{greeny}
\newlist{alternitem}{itemize}{1}
\setlist[alternitem]{label={\iftoggle{greeny}%
{\color{green!70!black}\faCommentingO\global\togglefalse{greeny}}%
{\color{blue!70!white}\faCommenting\global\toggletrue{greeny}}%
}}

\begin{document}

 \begin{alternitem}
 \item First person
 \item Second person
 \item First person
 \item First person
 \item Second person
 \item First person
 \end{alternitem}

\end{document}