You haven't said how you created the two site copies(db dump, export/import, etc), so...
Have you actually configured the local one to also use pretty URLs? (Settings->Permalinks) That wouldn't go along as part of a file copy; it's in the database. Or you might've just forgotten it.
If not, then your install is still expecting links in the default
If the local is a dupe of the published site, then try
http://localhost/?p=386 (that's the id of the contact page).
I haven't used WAMP for a bit so can't recall the case then, but apparently as recently as this February mod_rewrite wasn't enabled by default, which you'll need for the pretty URL rules to work. So that's one more thing to check.
Solution from the site mentioned above:
Wamp and XAMPP doesn’t come with default rewrite rule enabled, For enabling the rewrite rules in both of them you need to make changes in httpd.conf so that apache can work with rewrite rules.
For enabling the rewrite rules follow these rules:
Find httpd.conf (\wamp\bin\apache\Apache2.2.11\conf something along those lines
Uncomment the line LoadModule rewrite_module modules/mod_rewrite.so
Find the line with DocumentRoot “C:/path/to/my/root”, There you will find contents like
Make sure the content inside these two braces looks like
Restart the Apache server and you should be good to go.