How to format SI units?

inter-unit-product is an available option and described in the documentation. To get correct spacing you should use ${}\cdot{}$. However I prefer a tighter spacing by $\cdot$.

\documentclass{article}
\usepackage{siunitx}
\parindent0pt
\begin{document}
\SI[inter-unit-product =$\cdot$]{1.0}{\newton\meter}\\
\SI[inter-unit-product =${}\cdot{}$]{1.0}{\newton\meter}\\
$1.0\,\mathrm{N}\cdot\mathrm{m}$
\end{document}

To set his behavior for the whole document you can specify the option via \sisetup

\sisetup{inter-unit-product =$\cdot$}

Sorry for the zombie reply, but aren't the units typeset in math mode? In that case:

\SI[inter-unit-product =$\cdot$]{1.0}{\newton\meter}

won't compile. You should use:

\SI[inter-unit-product =\ensuremath{\cdot}]{1.0}{\newton\meter}

instead.