Unfortunately it seems that many folks here are a bit deluded about what being a Mashgiach in a food establishment entails. And a pizza shop is just as much liable for trouble. A Mashgiach must check every order and delivery as it comes into the store to make sure that a) only accepted items are ordered and that b) only those items are delivered. In a fleishig shop the meat needs to be checked, but cheese in a milchig place needs to be watched as well. Bugs, eggs, ovens, and employee use of equipment are only some of the jobs that they do.
The question of who pays the mashgiach is a problem of “eidus” in terms of shoichad which can pose a conflict of interest. If you have a question about a particulat hechsher or establishment, call the main office of that hechsher. Any skirting the of issue will indicate a problem.