Should job postings on bayt website be also posted on facebook page to promulgate job postings to a widespread scale using social media?

Most certainly, in fact it should not be limited to that. Given that a huge userbase on Twitter is from the Arab world ask for twisumes on twitter as well.

On Facebook, businesses should make use of Facebook Tabs to post jobs. Facebook restricts users from posting jobs on its ads (right panel Ad boxes).

jobs posting on social media are not considered authentic by majority of people

Agreed with Ahmad yar khan. Though, most of the job seekers tend to spend their time on social networking websites apart from job applications or may be when they're tired of applying. I guess a post on social media will enlighten the job over some different platform which will compel the applicant somewhat. Selection of the right candidate is up to the employers. 

social media is best for advertising as people are least interested to go for a detail about a job.

secondly, cannot connect with a right person at a right time , for that linkedln is best.