To buy a SIM card in a store, you have to show a photo ID (and your visa) in a store of the operators. In a tightening of laws you are expected to give your fingerprints as well from 2015. Note, that in public places you will be offered SIM cards by street vendors that are registered on someone's name. This should only be used as a last resort, as this is illegal and these cards can easily be switched off when detected.
In 2016 the regulator has started disconnecting unregistered mobile subscribers, which failed to submit their fingerprints in line with the new security measures introduced in September 2015. The regulator said the fingerprint registration is meant to protect personal information of SIM cardholders and prevent buyers from obtaining mobile phones using fake or stolen ID cards. All unregistered users will have a grace period of two weeks to submit their details, before the service is cut.
http://prepaid-data-sim-card.wikia.com/wiki/Saudi_Arabia
In 2016 the regulator has started disconnecting unregistered mobile subscribers, which failed to submit their fingerprints in line with the new security measures introduced in September 2015. The regulator said the fingerprint registration is meant to protect personal information of SIM cardholders and prevent buyers from obtaining mobile phones using fake or stolen ID cards. All unregistered users will have a grace period of two weeks to submit their details, before the service is cut.