Yes. When one is logged in as a gmail user, any google search is done as that user. Therefore the requests will carry session information just as if one were using gmail with http.
The authentication cookies for GMail are named "S" and "GX". They're both set only on "mail.google.com", and they both have the secure flag so they're only transmitted over SSL.
There are certainly other Google cookies that let them track your identity that will get sent when you search. But those cookies are not sufficient to get access to GMail.
