As the question states, is it allowed to bring snus (the Swedish tobacco pouches) into Saudi Arabia for own use when flying in? Note that I am not interested in knowing if it is possible, but only if it is allowed.
As stated by the IATA travel centre, which gets its information directly from local authorities:
Free import of:
(irrespective of age) 600 cigarettes or 100 cigars or 500 grams of tobacco (any excess prohibited);
a reasonable quantity of perfume or Eau de Cologne, for personal use;
a reasonable amount of cultured pearls, for personal use
So unless you exceed 500 grammes, you will be fine
According to my experience it is no problem to import snus, I use Siberia Snus myself, and it works very well! Sometimes they want me to pay VAT, sometimes not.
As someone else wrote above, it differs between officials when you ask. I also buy LYFT nicotine pouches in Saudi Arabia and sometimes also in Dubai, never any issue of those tobacco free products either.