For arriving at our hometown airport, both passports are with our boarding passes in my purse.....and I usually check to make sure they are there about 5 times before leaving, about three times en route to the airport, etc........yes, way too many times....nervous energy, I guess.....LOL.
Then, they are either hand held or put in my husband's shirt pocket until we go thru security, then they typically go back into my purse with the reprinted boarding passes....sometimes his will stay in his shirt pocket under his sports coat, if we fear we will get separated (last-minute run to the restroom and they start loading the plane, etc.).
But, once we are in our seats on the plane, they go into our individual RS neck-style money belts to remain there until we land and go thru immigration. We do not worry about accessing the neck wallets in public in the airport....I'd much rather someone see me doing that than sticking passports in my purse.
My husband wears an additional different neck-style money belt that I have adapted (shorted the string permanently) for holster style under his arm.....deep storage of extra cash, important numbers, etc. A certain portion of more accessible cash goes in the neck wallet, along with most-likely-to-use credit card. Then in his pocket and in the outer zip portion of my travel purse (and that zip remains toward my body) is what we call the daily cash. If it is hot weather, such that I'm not wearing a jacket or a top that is conducive to wearing the neck-style wallet, I will wear the waist style one (but I hate even that tiny bit of bulk at my waist).
The key thing about carrying one's passport/boarding tickets in the airport is to put them in your pocket or purse if you go to use the restroom. When I used to travel a lot on business, there were countless times I found tickets/passports sitting on the toilet roll holder in restrooms. I would take them to the gate (for the departing flight) and the agent would page the person. Not everyone would take that initiative (or may not seriously have the time to do so).
Planning for your trip, even the small questions, is good to switch your mind to fun topics after all you have been thru in that last month or so.
Oh, we also each keep a photocopy of the other's passport in our neck money pouch.