HELP - what is the salary of an ESL private teacher in Riyadh?

They haven't hired you as a ESL teacher, my guess is they have hired you as a nanny since it is a family that hired you versus a school.  Are you living with them??? And that would be the rate most families pay their maids/nannies per month.

I have been offered 13,000 SAR a month to teach English but that's for a public school.

Hope you get an idea!

The salaries being offered majorly depends on your nationality and then experience specially if it is related to English Language Teaching. The people from US, UK and other strong western countries gets higher pay scale as compared to others doing the same job.

My understanding is it depends on the specific requirement or personal preferences. For English Language instructor, being Native with a famous /familiar accent is a necessity most times. To achieve this, institutions are ready to pay as much as their budget stretch.

My 2 cents...

Khurram.

They pay according to your Passport.  Not skills.  2,000/mo = Bangladeshi.  2,800/no  = Filipino, Indian, Pakistani. 
  3,500 = Jordanian, Syrian
4,500/mo = South African   
  6,000/mo = American or British
   (These estimates are for Bachelors degree- MA slightly higher)

Dont even think of taking up this job. you will end up being a nanny or in worst case a slave, without any access to the outside world. Even house workers/nannies are contracted on a minimum 2500 riyals, and 2000 is no way enough to take care of your own expenses so I guess you will save more in your country, Do no believe in the agents and the promises they make. Not at all a good idea, instead apply at standard universities, you may get a good offer that way.