Cookies are intended to be specific to each machine, I don't think they get synced at all.
Chrome does however sync the passwords so it can automatically fill the log in details for a website the first time you use it on your phone, assuming you have previously logged in to the same site on (say) a PC. It's then up to you to check the usual "keep me logged in" box to set the appropriate cookie to keep you logged in on the phone.
The sync works both ways of course and includes for example the ability to reopen a site on one device that you recently visited on another plus lots of other useful features apart from just syncing the obvious things.
