Yes, it is true, Saudi Arabia has public executions, usually by beheading. There used to be a film clip on MEMRI which was of a tv interview with a headsman as he discussed his ‘craft’ and displayed his swords and techniques. It might still be there, but it would be several years old by now.
Another punishment which is public is be-handing for theft.
My roommate in high school personally witnessed one of these. She was around 12 when her father took her into the nearest Arab town with him. He parked in the town square and told her to remain in the vehicle while he went into the shop.
While she was sitting there, she heard a crowd approach, dragging a man who was yelling and screaming. There was a platform in the middle of the square. They dragged him onto the platform, placed his right arm on the block and, as she looked on in horror, chopped off his hand with a sword. Then they dipped his stump in boiling tar to cauterize the wound so he wouldn’t bleed to death. She said he screamed worse for the tar than for the removal of his hand.