I think you are mixing terms - in SharePoint land this would be an "anonymous"
site. Check out the administrator documentation for enabling anonymous access.
The process is different depending on which version you are running (STS, WSS,
SPS 2001, or SPS 2003).

About anonymous access

To allow users to contribute to a SharePoint site, list, or library
anonymously, you can configure the site  to allow anonymous access.
Anonymous access  allows users to browse sites, respond to surveys, or
even contribute to a list without authenticating. However, anonymous
users cannot add files to libraries.

To enable anonymous access in Microsoft Windows SharePoint Services, you
must first ensure that Internet Information Services (IIS) is configured
to allow anonymous access. After you configure IIS, you can enable
anonymous access on the Site Administration pages for the Web site, or
from the Change Permissions page for a list or library. To change
anonymous access settings in IIS, you must be logged on as a member of
the local administrators group on the server. To change anonymous access
settings in Windows SharePoint Services, you must be a member of the
Administrator site group, or a site group with the Manage Site Groups
right.

Anonymous access relies on the anonymous user account on the Web server
<http://buf-tp2:20129/help/sps/html/spsglos.htm> . Internet Information
Services (IIS) <http://buf-tp2:20129/help/sps/html/spsglos.htm> creates
and maintains this account, not Microsoft Windows SharePoint Services.
By default in IIS the anonymous user account is IUSR_ComputerName. When
you enable anonymous access in Windows SharePoint Services, you are in
effect granting that account access to the SharePoint site.

Warning To create  a more secure site, list, or library, do not enable
anonymous access. Enabling anonymous access allows users to contribute
to lists, discussions, and surveys; possibly using up server disk space
and other resources. Further, it allows anonymous users to discover site
information, including user e-mail addresses and any content posted to
lists, and libraries, and discussions.

If you set up anonymous access, anyone should be able to access the
site. Of course you should have some sort of Firewall and only allow
access on the port that your virtual server is configured to use. You
would not want it wide open.

If you want to have a private sharepoint  team site/ I.E. password protected
sharepoint enabled etc.and a public  site with no password access  necessary,
does IIs 6.0 and or sharepoint services allow this? Is it merely a question
of setting up permissions on two separate virtual servers?

Can it be done on the same server box having different settings in
IIS 6.0? example: (www.myteamsite.mydomain.com" target="_blank" rel="nofollow">www.myteamsite.mydomain.com
<http://www.myteamsite.mydomain.com" target="_blank" rel="nofollow">www.myteamsite.mydomain.com/> ) and (www.mydomain.com" target="_blank" rel="nofollow">www.mydomain.com
<http://www.mydomain.com" target="_blank" rel="nofollow">www.mydomain.com/> )

Right now I have a sharepoint teamsite for my intranet
(www.myteamsite.mydomain.com" target="_blank" rel="nofollow">www.myteamsite.mydomain.com <http://www.myteamsite.mydomain.com" target="_blank" rel="nofollow">www.myteamsite.mydomain.com/> ) and a
"static" site  for my public site (www.mydomain.com" target="_blank" rel="nofollow">www.mydomain.com
<http://www.mydomain.com" target="_blank" rel="nofollow">www.mydomain.com/> ) residing on the same box. I was wondering if
it were possible to have my public site using sharepoint extensions on the
same box.

Hope that made sense, I'm a bit confused about it myself.

Anyway, my old public site used to run on fp 2002 extensions and doesn't
seem to have any extensions now, 2002 or sharepoint so my form pages are
almost useless (they don't email out to me anymore). I have to check them
manually for new data and that is a pain. If I could extend the sharepoint
extensions to the old site, maybe I could set up alerts to new form data.

Without of course compromising security............

Just to avoid confusion, we are running on a WSS and
Sharepoint Portal 2003 on a Dev Windows 2003 Box connected to a SQL 2000
Server on Windows 2000. There are two of us working with it and we still
have a lot to learn. It seems to have some quirks, but at least you have
the option to do some development.

If both sites were to be accessible from the internet, you
could host both sites on the same box using different virtual sites and
host headers. At a very high level, one virtual site  (www.mydomain.com
<http://www.mydomain.com/> ) would have anonymous access  and would not
need a host header. It would probably not be using SSL if it is
anonymous. The second virtual site could use the and have a host header
of "myteamsite.mydomain.com" and the would accept all traffic from that
url. You would probably want to make this a secure site using SSL.

Of course, if you wanted your intranet site to remain as
inaccessible as possible from the web, you would not want the public
site and the intranet site on the same box. They should be physically
separated and depending on the level of security you wish to attain, you
might also want to have the public  server in a DMZ or unable to
communicate to the intranet site/server.

There is quite a bit you could do, it just depends on your goals. As
Bryan had mentioned, the best place to start is the help  files in the
version you are using.