By default paragraphs are numbered within subsections, restarting at 1 at each subsection. If you just wish to changed the printed representation of the paragraph number then all you need is
\renewcommand{\theparagraph}{\S\arabic{paragraph}}
\setcounter{secnumdepth}{4}
If you wish the paragraphs to also be numbered consecutively throughout the document then you can use the chngcntr package's command \counterwithout to undo the resetting as follows:

\documentclass[12pt]{article}
\usepackage{chngcntr}
\counterwithout{paragraph}{subsubsection}
\renewcommand{\theparagraph}{\S\arabic{paragraph}}
\setcounter{secnumdepth}{4}
\begin{document}
\section{First section}
\paragraph{}\label{Personal Statement}
San Francisco is a city in California.
\subsection{A subsection}
\paragraph{}
San Francisco is a city in California. See paragraph \ref{Personal
Statement}.
\subsubsection{A subsubsection}
\paragraph{}
San Francisco is a city in California.
\section{Second section}
\paragraph{}
San Francisco is a city in California. See paragraph \ref{Personal Statement}.
\paragraph{}
San Francisco is a city in California.
\end{document}
Added in response to comment To adjust the spacing after numbers etc. it is probably easiest to use the titlesec package. There are two commands involved here: \titleformat and \titlespacing. If you never intend to give titles to your paragraphs then the \titlespacing command below is sufficient. The \titleformat command makes sure words in a paragraph heading are also handled appropriately:

\documentclass[12pt]{article}
\usepackage{chngcntr}
\counterwithout{paragraph}{subsubsection}
\renewcommand{\theparagraph}{\S\arabic{paragraph}}
\setcounter{secnumdepth}{4}
\usepackage{titlesec}
\titleformat{\paragraph}[runin]{\normalfont\bfseries}{\theparagraph}{\wordsep}{}
\titlespacing{\paragraph}{0pt}{3.25ex plus 1ex minus .2ex}{\wordsep}
\begin{document}
\section{First section}
\paragraph{}\label{Personal Statement}
San Francisco is a city in California.
\subsection{A subsection}
\paragraph{}
San Francisco is a city in California. See paragraph \ref{Personal
Statement}.
\subsubsection{A subsubsection}
\paragraph{}
San Francisco is a city in California.
\section{Second section}
\paragraph{}
San Francisco is a city in California. See paragraph \ref{Personal Statement}.
\paragraph{With title}
San Francisco is a city in California.
\end{document}
The arguments given to \titlespacing are as follows:
- section command =
\paragraph
- left indent =
0pt
- above vertical skip =
3.25ex plus 1ex minus .2ex the standard value from the article class
- final separation =
\wordsep an ordinary word space
For \titleformat the spcification is
- section command =
\paragraph
- style =
runin for heads contain in the same paragraph as the text
- format =
\normalfont\bfseries for styling the heading
- label =
\theparagraph the printed representation of the section number
- sep =
\wordsep is the space between the label and the title (when present)
- before code = empty
\renewcommand{\theparagraph}{\S~\arabic{paragraph}}– karlkoeller Aug 05 '13 at 06:01