Would this work for you? I just use the description list.
Version 1
\documentclass{article}
\usepackage{blindtext}
\usepackage{paracol}
\usepackage{parskip}
\begin{document}
\section*{Random Text}
\begin{paracol}{2}
\begin{description}
\item[Person 1] \blindtext
\end{description}
\switchcolumn
\begin{description}
\item[Person 2] \blindtext
\end{description}
\end{paracol}
\end{document}

Version 2
Here I added the enumitem and the calc package.
\documentclass{article}
\usepackage{blindtext}
\usepackage{paracol}
\usepackage{parskip}
%% Important here
% http://tex.stackexchange.com/questions/9760
% http://tex.stackexchange.com/questions/7008
% http://tex.stackexchange.com/questions/217722
% http://tex.stackexchange.com/questions/33817 <-- very relevant
\usepackage{calc}
\usepackage{enumitem}
\setlist[description]{
leftmargin=18mm,
itemindent=0mm,
labelwidth=\leftmargin-\labelsep
}
\begin{document}
\section*{Random Text}
\begin{paracol}{2}
\begin{description}
\item[Person 1] \blindtext
\end{description}
\switchcolumn
\begin{description}
\item[Person 2] \blindtext
\end{description}
\end{paracol}
\end{document}

Version 3 -- Best solution in my Opinion
\documentclass{article}
\usepackage{blindtext}
\usepackage{paracol}
\usepackage{parskip}
\usepackage{calc}
%% Important here
% http://tex.stackexchange.com/questions/9760
% http://tex.stackexchange.com/questions/7008
% http://tex.stackexchange.com/questions/217722
% http://tex.stackexchange.com/questions/33817 <-- very relevant
\usepackage{enumitem}
\setlist[description]{
leftmargin=\widthof{\normalsize\textbf{Person 99}},
style=nextline
}
\begin{document}
\section*{Random Text}
\begin{paracol}{2}
\begin{description}
\item[Person 1] \blindtext
\end{description}
\switchcolumn
\begin{description}
\item[Person 2] \blindtext
\end{description}
\end{paracol}
\end{document}