Welcome to TeX.SX! You could make use of the cuted
package like in the following example. It may be possible, I think, that LaTeX doesn't always prevent page breaks between the section title and the following paragraph. Maybe some adjustment is needed.
\documentclass[twocolumn=true]{scrbook}
\usepackage{lipsum, cuted}
\renewcommand{\sectionlinesformat}[4]{%
\begin{strip}\thesection\autodot\enskip #4\end{strip}%
}
\begin{document}
\chapter{Chapter 1}
\section{Section 1}
\lipsum[1-3]
\section{Section 2}
\lipsum[3-6]
\section{Section 3}
\lipsum[1-4]
\chapter{Chapter 2}
\section{Section 1}
\lipsum[1-2]
\section{Section 2}
\lipsum[3-4]
\section{Section 3}
\lipsum[1-5]
\end{document}
First page of the output:

Edit: If you don't want to use the same styling for subsection headings, you can use the following to apply the above settings only to sections. I also changed the code so that the starred version of section title won't get a section number:
\documentclass[twocolumn=true]{scrbook}
\usepackage{lipsum, cuted}
\makeatletter
\renewcommand{\sectionlinesformat}[4]{%
\Ifstr{#1}{section}{%
\begin{strip}\@hangfrom{\hskip #2#3}{#4}\end{strip}%
}{%
\@hangfrom{\hskip #2#3}{#4}%
}%
}
\makeatother
\begin{document}
\chapter{Chapter 2}
\section{Section 1}
\lipsum[1-2]
\subsection{Subsection 1}
\lipsum[2]
\section{Section 2}
\lipsum[3-4]
\section{Section 3}
\lipsum[1-5]
\end{document}
The first page of the output renders like this:
