Slavery in Saudi Arabia is a difficult topic to tackle, mainly because of scope and definitions. (I know more about slavery in the 70's with wealthy families, so anyone who knows more on the 50s and general slavery can correct me if I'm wrong)
There are so many kinds of workers in very difficult conditions, so it's difficult to define what is what before talking about it without generalizing.
There are families (usually wealthy) who have their own personal slaves. The slaves are descendents of slaves and have been with the families for generations. They might have no (known) connection to their homelands in Africa anymore. They are born into slavery. They may not have Saudi citizenship (despite being born there), almost certainly no Saudi passport and no other citizenship from another country. They are most likely stateless. Wealthy families employ them in domestic jobs - cooking, cleaning, babysitting, gardening, driving the women/kids, bodyguards and so on. These slaves are not traded. The families would have known the slave since s/he was a baby and would not trust someone new into their homes, with their women and kids. The slaves travel with the Arab families and perform these same duties abroad. There have been reported instances of the Qatari royal family traveling on diplomatic missions with their slaves to the United States and Britain, but even ambassadors do that too (bring slaves to the new country) and this is done even by non-royal families.
There was regular slavery. Slaves traded in markets. These slaves have a connection to their African homelands but could not go back because of conflict in their countries, poverty or because of exit-visa restrictions. They were not tied to a specific family (not born into slavery) but they cannot leave. I don't know much about this type, but there were some poor families that owned slaves. They were used to do domestic work at home and to drive the women and kids. I don't know if large-scale slave labor existed or how it was like. I hope someone will add to this.
There are also a lot of abuses in employment in Saudi Arabia - from employers not paying their staff and not letting them quit or leave the country, or paying them very little for a lot of work, letting them live in extreme poverty (which is made worse by the workers sending most of their money back home), seizing their passports, hitting and beating them and so on. Immigration and employment laws put migrant workers at a disadvantage, tying them to the company they're employed at and making it very difficult to change jobs or leave the country. Sometimes poverty in their home countries makes it difficult to leave, preferring to endure the miserable conditions, but other times employers take the extra steps of seizing their passports, threatening to report them for crimes (stealing from the employer) which would get them arrested at the airport and imprisoned. Most of these workers work in construction - they don't live with their employers, they rarely do domestic jobs, they are not traded and are not considered slaves (not by language, law or culture).
The lines between all these groups are blurry. Is this last group slavery or not? If a wealthy slave owner pays his slaves, treats them well, their quality of life is better than most people (including other poor slave owners), and gives them the permission to travel/leave, but they are stateless people who have no right to leave the country or work elsewhere, is it still slavery? Let's say "yes" and now slavery is abolished, where would these slaves go? They are still stateless and have no right to work elsewhere.
The issue is not a single problem (the legality of slavery), but a multitude of other problems such as citizenship and immigration, exit-visas, employment rights and economics that all work together to put certain groups at a disadvantage.
Note: Some slaves did have citizenship and passports. There was a time when slave passports would list the owner of the slave on the (slave's) passport, but this was removed (I think due to diplomatic pressure.)
Listing the owner's name may have prevented the slaves from traveling without their owners' permission (border/airport security would check), but I don't know how this worked when this part was removed from the passports. Slaves traveled with diplomatic missions and regular slave owners, but I don't know of any reports of slaves making a run for it once they were outside.