Text before maketitle

Don't specify the twocolumn document class option. Instead, load the multicol package and execute the command \begin{multicols}{2} after \maketitle -- as well as \end{multicols} before \end{document}, of course. That way, you can typeset various things before the title stuff gets typeset with the \maketitle command.

The following, simplified, version of your MWE illustrates how this may be done.

\documentclass[a4paper,twoside]{article}
\usepackage[utf8]{inputenc}
\usepackage[MeX]{polski}

\usepackage{multicol,lipsum}
\title{Tytul}       
\author{Iksinski}
\date{some date}

\begin{document}
\noindent
left-hand text \hspace{\fill} right-hand text\\
more left-hand text \hspace{\fill} more right-hand text

\begingroup
\let\newpage\relax
\maketitle
\endgroup

\begin{multicols}{2}
\lipsum[1] % filler text
\end{multicols}
\end{document}

enter image description here


Based on the image you provided, you need to create a custom header that includes some content. The easiest way to do this, depending on your requirements or limitations, would be to use the fancyhdr package. For example, creating a "title" header you could use:

enter image description here

\documentclass[twocolumn]{article}
\usepackage{lipsum}% http://ctan.org/pkg/lipsum
\usepackage{fancyhdr}% http://ctan.org/pkg/fancyhdr
\fancypagestyle{title}{%
  \setlength{\headheight}{22pt}%
  \fancyhf{}% No header/footer
  \renewcommand{\headrulewidth}{0pt}% No header rule
  \renewcommand{\footrulewidth}{0pt}% No footer rule
  \fancyfoot[C]{\thepage}% Page number in Centre of footer
  \fancyhead[L]{\small\begin{tabular}{@{}l}Psycological Assessment​​​​​​​​​​​​​​​​​​​​​​\\2011, Vol.\ 23​​​​​​​, No.\ 1, 64--79\end{tabular}}
  \fancyhead[R]{\small\begin{tabular}{@{}r}\copyright~2001​​​​​​​​​​​​​​​​​​​​​​​​​​​ American Psycological Association\\1040-1234567890~XYZ:\ 8472302\end{tabular}}
}%
\title{Here is my title}
\author{Some Author}
\date{\today}
\begin{document}
\maketitle
\thispagestyle{title}

​\lipsum
\end{document}​​​​​

If your document class provides an interface to the headers (for conference or paper/article submissions, say), it's best to use that.

Read the fancyhdr documentation for more information on the commands used and how to modify the respective header/footer locations/contents.

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