BRASOV, Romania—Like thousands of cops the world over, Marian Godina got an angry call from his bosses. They wanted to see him immediately.

Some years earlier Officer Godina, a traffic officer in this city of 250,000 in the mountains of Transylvania, had pulled over a drunken driver who, in a break from the ordinary, had chosen to acknowledge his inebriation—with gusto. Asked if he had been drinking, the driver replied: “Do I look sober to you?” Asked what types of drinks he’d had, he said: “All sorts. Write down, a-l-l s-o-r-t-s.” When asked if he knew what day it was, he said: “It’s night…not day,” tossing in a few choice expletives.

When Officer Godina’s bosses saw, last summer, that he had posted the incident on Facebook, they weren’t amused by the coarse language. They ordered him to take down his entire Facebook page. Officer Godina, now 29 years old, was so humiliated he thought about quitting the force.

But overnight, after his page went dark, the Brasov police server crashed. It had been bombarded with complaints in his support. The next day, his bosses reversed course and ordered him to reactivate the page.

Though they hadn’t known it, Officer Godina’s Facebook page had become a cause célèbre. For decades, Romanian traffic police officers had been the subject of endless jokes about their alleged lack of integrity, education and wit. Officer Godina’s page was so relentlessly honest, funny, grammatically correct and at times openly critical of the corruption he saw firsthand that he had developed a loyal following among fed-up citizens.