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When they are riding on the jumpseat, they are observing your operations - if they have any questions - they should only ask/speak during non-pertinent times of the flight - and if they become a distraction - the PIC should remind them they are only observing.
They have the right to use the JS for personal travel whenever they want and whenever the loads are such that they can't visit sons, daughters, grandchildren, beach, for free without giving you a "line check" to get there.
Sorry to be the nerd in the group ... there is such a document (actually it is guidance and a Job Aid (FAA speak for a checklist)).
FSIMS (Flight Standards Information Management System) / Volume 8900.01 / Volume 6 / Chapter 2 / Section 6 and 9. I would copy both sections but I would be banned for life for exceeding the character limits on a post.
In short, an Inspector may ask any question(s) they want to determine whether the flight crew and the operations meets the regulatory requirements of FAR121 and the approved OPSPEC / FOM.
The inspectors who have ridden on the J/S at SWA have been cordial and professional. Never once have they been 'in the way' or a distraction. The FAA at the ASI level is pretty good at policing their own.
If you read the above mentioned guidance you will see the word OBSERVE used over and over again. If there is an issue which could result in 'paperwork' or 'time off' they are obligated to inform the captain of such.