OK, I feel pretty silly. Just seconds after posting I found:
https://www.census.gov/popclock/data/population.php/us
https://www.census.gov/popclock/data/population.php/world
which gives replies like this:
{"us":{"label":"United States","estimate":true,"population":323177593,"population_rate":0.067974537,"rate_interval":"second","last_updated":"1458140658","components":{"birth_rate":{"increment":1,"interval":"8","unit":"seconds"},"death_rate":{"increment":1,"interval":"11","unit":"seconds"},"immigrant_rate":{"increment":1,"interval":"29","unit":"seconds"}},"date":"03\/16\/2016"}}
{"world":{"label":"World Population","estimate":true,"population":7312020325,"population_rate":2.47551113,"rate_interval":"second","last_updated":"1458140658","monthly_estimates":{"1435708800":{"population":7256490011,"estimate":true,"projected":false},"1438387200":{"population":7263120419,"estimate":true,"projected":false},"1441065600":{"population":7269750827,"estimate":true,"projected":false},"1443657600":{"population":7276167351,"estimate":true,"projected":false},"1446336000":{"population":7282797759,"estimate":true,"projected":false},"1448928000":{"population":7289214283,"estimate":true,"projected":false},"1451606400":{"population":7295844691,"estimate":true,"projected":false},"1454284800":{"population":7302475099,"estimate":true,"projected":false},"1456790400":{"population":7308677739,"estimate":true,"projected":false},"1459468800":{"population":7315308147,"estimate":true,"projected":true},"1462060800":{"population":7321724671,"estimate":true,"projected":true},"1467331200":{"population":7334771614,"estimate":true,"projected":true}},"date":"20160316"}}
So, yes, the clock does contact a server everytime you load the page.