They carry out beheadings a lot more frequently than the SAA does, and when it takes place in Saudi Arabia its not due to a few rogue officers but rather due to official ruling. Same with ISIS. I think you'd be wise to distinguish between what ISIS and Saudi Arabia carry out and what some SAA on the front lines carry out.

As for the beheadings being public, they most certainly are. They do occasionally carry out private beheadings, I'll grant you that. That doesn't mean they don't carry out public ones. I'm sure there are many things in this world that you and your friends haven't personally seen, but that doesn't mean they don't take place.

Here you go, because science:

http://www.liveleak.com/view?i=3fe_1335007843

http://www.liveleak.com/view?i=0c5_1320328069

http://en.wikipedia.org/wiki/Public_executions_in_Saudi_Arabia

A public beheading will typically take place around 9am. The convicted person is walked into the square and kneels in front of the executioner. The executioner uses a sword known as a sulthan to remove the condemned person's head from his or her body at the neck. Sometimes it may take several strikes before victim is decapitated. After the criminal is pronounced dead, a loudspeaker announces the crimes committed by the beheaded alleged criminal and the process is complete. This is the most common method of execution in Saudi Arabia because it is specifically called for by Sharia Law. Professional executioners behead as many as ten people in a single day. The severed head is usually sewn back on, and sometimes put on crucifixes for public display. In 2011, an Indonesian maid’s dead body was hung from a helicopter for display.

http://en.wikipedia.org/wiki/Capital_punishment_in_Saudi_Arabia

Unlike executions in most other countries that have not abolished the death penalty, executions of offenders are not performed privately in prisons, but publicly in central Riyadh, and have been called the "only form of public entertainment" in Saudi Arabia "apart from football matches". It is one of the last four countries to still carry out public executions and the only country to carry them on regularly.