I'm just SLF but I can tell you that Exit Permits apply to everyone working (legally) in Qatar - not just employees of State organisations. The private sector is hit too.
AFAIK the only other countries which apply similar restrictions are K.S.A. and Iran.
Good employers can get them instantly via an on-line process. It's no big deal even if you don't have a multiple exit permit. The key words are 'good employers'.....