Edit after comment: I think I finally understand what is the question you actually want to answer.
I use fancyhdr to have chapter and section titles in the header. However, for the pages of the table of contents, the chapter title "CONTENTS" appears on both sides of the header, i.e. also as the section title. How can I remove it from the one side of the header?
This has a simple answer: Use \markright{} to clear the "section title" part of the header. You can do so by wrapping it into \addtocontents{toc} (to have it right after the chapter title calls \markboth on the first page of the contents) and \AtBeginDocument (to have it at the first "entry" in the table of contents).
\documentclass[a4paper,11pt]{book}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{fancyhdr}
\usepackage{lipsum}% used to get dummy text
\AtBeginDocument{\addtocontents{toc}{\protect\markboth{\textsl{CONTENTS}}{}}}
\begin{document}
\pagestyle{fancy}
\tableofcontents%
\cleardoublepage
\chapter{Chapter 1}
\section{Section 1.1}
\lipsum[1]% used to get dummy text
\cleardoublepage
\chapter{Chapter 2}
\section{Section 2.1}
\lipsum[2]% used to get dummy text
\chapter{Chapter 3}
\section{Section 3.1}
\lipsum[3]% used to get dummy text
\end{document}
Header from page 2: 
Old answer:
Without any manual intervention, the default header at the subsequent pages of the table of contents has "CONTENTS" in captial letters both left and right. The first page of the table of contents doesn't have any header, as all other opening pages of chapters.
\documentclass[a4paper,11pt]{book}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{fancyhdr}
\usepackage{lipsum}% used to get dummy text
\begin{document}
\pagestyle{fancy}
\tableofcontents%
\cleardoublepage
\chapter{Chapter 1}
\section{Section 1.1}
\lipsum[1]% used to get dummy text
\cleardoublepage
\chapter{Chapter 2}
\section{Section 2.1}
\lipsum[2]% used to get dummy text
\chapter{Chapter 3}
\section{Section 3.1}
\lipsum[3]% used to get dummy text
\end{document}
Header from page 2: 
To modify what is put into the header for the pages of the table of contents, you have to call \markboth which overwrites what has already been put there by the caption of the table of contents. There is no need to hack any other commands to not call \markboth or related commands, as the purpose of these macros is really to overwrite what is already there. So you really only need to call \markboth after the chapter heading of the table of contents and before the first page break, i.e. ideally directly after the chapter heading.
Inspired by this answer, I would add the \markboth call to the TOC as first entry (so that it's called directly after the chapter heading). To make sure that it's the first entry, you can use \AtBeginDocument in the preamble. Minimal example:
\documentclass[a4paper,11pt]{book}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{fancyhdr}
\usepackage{lipsum}% used to get dummy text
\AtBeginDocument{\addtocontents{toc}{\protect\markboth{Contents}{\textnormal{More Contents}}}}
\begin{document}
\pagestyle{fancy}
\tableofcontents%
\cleardoublepage
\chapter{Chapter 1}
\section{Section 1.1}
\lipsum[1]% used to get dummy text
\cleardoublepage
\chapter{Chapter 2}
\section{Section 2.1}
\lipsum[2]% used to get dummy text
\chapter{Chapter 3}
\section{Section 3.1}
\lipsum[3]% used to get dummy text
\end{document}
Header from page 2: 
You will of course now want to modify this, to have the exact wording and also text formatting style for the header that you need. Note that the default style (at least in this minimal example) is italicized text, so you may need to overwrite that.
\makeatletter ... \makeatotherpart, I get titles for the contents and the chapters on the respective first pages, and properly labelled headers on all subsequent pages. Maybe you can use a graphics program to sketch how the result should look like? – Tiuri Dec 02 '19 at 14:49