I believe that the notice would be intrepreted and applied as if it read:
By continuing to use our site, you acknowledge that you have read and understand our Cookie Policy, Privacy Policy, and our Terms of Service.
In short, if after having seen the notice (or the page on which the notice appears) you continue to use the site, you will be assumed to have read those policies. Whether such a notice will be effective will depend on the jurisdiction, and the circumstances.
For example, if the notice is in smaller type than the rest of the page, it may not be considered reasonable-to assume that a user has read it. If the user would have to scroll down to see the notice, but could use the page and follow links to other parts of the site without scrolling down it again may not be given effect by a court, should the matter come to litigation.
Better practice would require the user to positively acknowledge and accept the terms, with a button click or a check box or some similar mechanism, before allowing access to the rest of the site.