Checkins can easily be faked. The Foursquare app does a reasonable job of preventing fake (and accidental fake) checkins, but it's still possible. If checkins weren't posted on Foursquare, but instead were posted on each person's own website, the possibility of fake checkins is much greater. What would it look like to have a way for a venue to know (and republish) checkins that it knows were real?
The checkin post would need to include some piece of information that could only have been discovered by physically being at the venue.
The venue's TV screen always displays a 4-digit code and instructs people to include that code in their checkin post on their website.
When someone posts a checkin on their site, they link to the venue URL, so their site sends a Webmention to the venue's website.
The venue's Webmention receiver then looks at the checkin post, and verifies that the special code is included and that it has not already been used.
By providing people this code in the physical location, there is no way for someone to know the code unless they were actually there!