In order to do this, you need to create a separate format for the partial TOCs by using titletoc's prefix system.
Creating separate formatting for the partial TOCs
First we create the format for a psection instead of a section using the regular \titlecontents command. This can be formatted however you like. To make it flush with the left margin we make the left margin 2.3em (same width as the space set up for the section number). Then when we use the \printcontents command to produce the partial contents we use the {p} prefix argument to tell titletoc to use the format defined for psection instead of section. You need to determine the formatting for the subsections in the same way if you have subsections. I've removed some extra space that I added before.
Here's an example:
\documentclass[oneside]{book}
\usepackage{titletoc}
\usepackage{lipsum}
\setcounter{tocdepth}{2}
\begin{document}
\titlecontents{psection}[2.3em]
{} {\contentslabel{2.3em}} {} {\titlerule*[1pc]{.}\contentspage}
\titlecontents{psubsection}[5.5em]
{} {\contentslabel{3.2em}} {} {\titlerule*[1pc]{.}\contentspage}
\tableofcontents
\chapter{A chapter}
\startcontents[chapters]
\printcontents[chapters]{p}{1}{}
\section{Section}
\lipsum[1]
\subsection{A subsection}
\lipsum[3]
\subsection{A second subsection}
\lipsum[4]
\section{Section 2}
\lipsum
\chapter{Second chapter}
\startcontents[chapters]
\printcontents[chapters]{p}{1}{}
\section{Section}
\lipsum[2]
\section{Another section}
\lipsum
\end{document}

Choosing the correct spacing values
The values you choose to format the margins for the various levels can be set by hand. The default values used by the book class are the following. These values are those passed to the \@dottedtocline command within the internal sectioning command definitions. They were obtained from the source code of book.cls.
Level Indent Label width
section 1.5em 2.3em
subsection 3.8em 3.2em
subsubsection 7.0em 4.1em
paragraph 10.0em 5.0em
subparagraph 12.0em 6.0em
You can use these values to determine the values for the \titlecontents parameters for the various sections. Since our partial TOCs don't have a chapter level, we start the left margin at the width of the label for section (2.3em) and then use the same value for the label width. To make the subsections indent exactly the same amount as in the main TOC relative to the section, we make the left margin = previous left margin + current label width, i.e. 5.5em. Lower levels proceed similarly.
\quadI inserted. I've removed it. For subsections you need to add a separate\titlecontentsfor that too, otherwise it defaults to whatever is set for the main TOC. I've added that to the revised answer. – Alan Munn Aug 08 '12 at 17:31[2.3em]and\contentslabel{2.3em}in order to get the correct alignment for levelsubsubsectionsandparagraphsandtocdepth{4}. It would be therefore useful to know what the effect of[2.3em]and\contentslabel{2.3em}is and where one can look up the right settings for all ToC levels ...titlesecdocumentation. The left margin is where the entry wraps to, and doesn't include the space taken up by the label, so to get the label lined up you need to increase the left margin by the space taken up by the label. This value is additive, so for every subsequent level, you need to add the space taken up by the previous level as well. I don't know if there's a simple way to calculate this automatically. – Alan Munn Aug 08 '12 at 18:42bookuses in the main TOC. – Alan Munn Aug 08 '12 at 19:56