This information was posted on Facebook's Blog earlier this year with some merges and summaries made to fit your case.
What information is being shared with groupon.com?
None of your information—your name or profile information, what you like,
who your friends are, what they have
liked, what they recommend—is shared
with the sites you visit with a
plugin. Because they have given
Facebook this "real estate" on their
sites [At a technical level, social plugins work when external websites put an iframe from Facebook.com on their site], they do not receive or interact
with the information that is contained
or transmitted there. Similarly, no
personal information about your
actions is provided to advertisers on
Facebook.com or on the other site.
While these buttons and boxes appear
on other websites, the content
populating them comes directly from
Facebook. The plugins were designed so
that the website you are visiting
receives none of this information.
These plugins should be seen as an
extension of Facebook.
Is this sharing of information something I can disable in facebook?
You should consider the likes and
recommendations you choose to make to
be public information, much like when
you comment or write a review on any
website today or connect with a public
Facebook Page.
Depending on the connection you made,
you can remove a connection completely
either by going to the "Info" tab of
your profile and editing your "Likes
and Interests," or by returning to the
web page where you liked something and
unliking it.
Likes and Recommendations made on
other sites become publicly available
information, similar to a public
comment on a website. Only click the
Like or Recommend buttons if you want
to share your likes and
recommendations publicly.
In my Facebook Privacy Settings, "Enable instant personalization on
partner websites." is unchecked which
I thought prevented sites like Groupon
from getting my Facebook information.
What am I missing?
Only an exclusive set of
partners—Microsoft Docs.com, Pandora
and Yelp offer personalized
experiences as soon as you visit those
services. These partners have been
given access to public information on
Facebook—such as names, friend lists
and interests and likes—to personalize
your experience when you're logged
into Facebook and visit their sites.
When you first visit any of these
three partner sites while logged into
Facebook, you'll see a blue bar appear
at the top of the site letting you
know that your experience is being
personalized. You can learn more about
it, remove the personalized experience
or click "x" to remove the bar.
Best Answer
Only if your Facebook privacy settings are set to allow non-friends to see your whole profile. It is not affected by whether the other person Likes (is a fan of) the page. You can find out what your profile looks like to non-friends by going to Account » Privacy Settings » Basic Directory Information - View settings » Preview My Profile. (If you have permitted some information to be shown to friends-of-friends or networks then people in those categories will see the additional information.)
People will be able to click on your name or picture in your comment (if you haven't explicitly blocked that person), but they will see the same partial profile for you regardless of whether they Like the page. Of course, people who Like the page are more likely to see your comment but anyone could potentially visit the page. If you have turned off your visibility in search so that people cannot find you, someone that may not have been able to find you on Facebook previously may discover you when they see your comment.