I know that if I enter my own frequent flier number on another person's reservation/check-in, I will get miles for their flight. But let's say I have a certain status with the airline - would entering the frequent flier number grant the same privileges to the other passenger? Or perhaps get them higher on the upgrade list in case of an overbooking?
In general, the frequent flier number is only valid for that single passenger. On a multi-passenger itinerary, if the principal booking passenger has status with an airline, some privileges might extend to other passengers on the same booking (e.g. United will give an earlier boarding category to such passengers), but you'll only earn miles or points for yourself. Otherwise, status passengers could abuse their privilege and let almost anyone avail themselves of the status.
The only way to get other passengers benefits from a frequent flier number would be for them to have their own number (aside from the boarding priority you can pass along to traveling companions). Of course, in this case, they'll just be earning points for themselves.
There are some borderline cases. When you book a ticket with miles for someone else, and you're on a higher tier, the traveler might be treated a little better. I booked once a ticket on Air France for someone, which had a few segments. I was a Platinum member back then, and the person traveling on this ticket faced delays on one of the segments, potentially jeopardizing the rest of the flights, and AF staff went really out of their way to help. They mentioned that my status was partly the reason.
But on the other hand, I haven't seen yet airlines extend other benefits (lounge access, upgrades, etc) to people flying on tickets procured by higher-tier members.