By Drew Hinshaw, Francis X. Rocca and Natalia Ojewska 

WEGROW, Poland—One Sunday morning last year, the Rev. Jacek Wentczuk stood before his congregation and made a startling admission. He was a convicted sex offender, he said, found guilty in 2012 of molesting a 15-year-old girl in a nearby town. His bishop was considering transferring him.

Parishioners rallied to the side of the popular Catholic priest, who insisted he had done nothing wrong. They brought flowers and children’s drawings to persuade church leaders to keep Father Wentczuk in his job in this small town in eastern…