If you are a citizen you need to use that passport to enter and leave the country you are a citizen of. If not a citizen use the passport that Benefits you the most, but yes enter and leave on the same passport so it’s recorded in and out.
Note. You don’t have to show airline staff and passport control the same passport. When going from Switzerland to the UK; I show Swiss immigration my Swiss passport, then show the boarding agents my British passport.
Swiss nationals should show Swiss documents entering and leaving Schengen.
Most other countries require their nationals to use their documents entering and leaving their territory.
Not all countries have passport controls on exit. US, Canada, UK for example.
But in most cases airlines will want you to present them documents allowing you entry to the destination countries. So a Elbonian/Swiss joint national leaving Elbonia for Switzerland should show the airline the Swiss passport and the authorities the Elbonian one.
If it is airline company then you will need to show a passport that allows you to enter or have a valid visa or you will not be allowed to board the airplane.
If it is the customs officer, he doesn't really care as you are leaving, they only really check when they are looking for somebody when you leave a country as you are, theoretically no longer going to be their problem
As others have said, it depends. The rules of the country you are entering prevail in most cases in my experience. What passport you leave a country is typically not so important. What you show customs officials vs. what you show the transport carrier (usually airline) can also be different. I have seen a tightening of said rules and practices in recent years, some countries are definitely stepping up scrutiny.
For what it's worth, I have to enter USA on US passport as US citizen. They also now say have to use it on exit, although I'm yet to find the statute on that one. Last time leaving I had an issue because US passport was expiring in a matter of a few weeks which was not sufficient to enter Schengen (Switzerland). The airline blocked my online check-in and said I was not eligible to travel. So I had to delete all US passport details on my airline profile, replace with CH passport details. This got around the controls of the airline to check-in etc.
At the airport, and this depends on airport in my experience, some US airports have exit customs checks, others not, but seems it is being rolled out gradually. The customs wanted to see the US passport, not the one I was travelling on. This was only a visual check by officer, not scanned or registered in any way, but who knows how this procedure will evolve. So I left US on US passport for customs and Swiss passport for airline. Airlines have to verify the country entry requirements at destination so this makes sense.
In my personal experience of one (me!), it doesn't seem to matter. My UK passport expired and I travelled CH/UK just on my CH passport and I've also just travelled on my UK passport back and forth. Neither set of border agents batted an eye.
This. I carry both passports every time I travel, and I do what I call "shuffle and deal" depending on who needs to see what.
Even if staying within Shengen, I have my US passport with me. Life throws curves at us, and if I need to get to the USA at a moment's notice I'd best have that passport in my possession!
I have not renewed my Irish passport in a decade and I just use a Swiss ID card when entering Ireland. I was only stopped once by immigration and it was only a joke - I was travelling with an American who was using an Irish passport... the officer's comment was: "A Swiss person with a very Irish family name and an Irish person with a very Germanic family, nothing suspicious there at all" and he waved us true.
For the UK you're right, there is no requirement to use a UK passport if yo have another acceptable one. But Schengen does require Schengen nationals to enter/exit on their Schengen passport - although enforcement may depend on the individual passport control officer. Exited Schengen in Memmingen recently and they did insist.
Problem here, a third national will be entered into the Schengen information system and an overstay might be flagged. Obviously, it is not a real overstay as you are also a Schengen/EU citizen. Not the passport is relevant, but your actual citizenships.
However, I am not aware of a rule which says you must you user EU/Schengen passport, specially as your status as a person enjoying free movement is not determined by the passport you have or us, but by the citcenship you have.
Notably, there is no regulation that Swiss citizen must use Swiss ID card or passport when entering Switzerland. Once again, your status in the country is not determined by the passport you use, but by the citizenships you hold.
This also means a dual citizen is not here as either citizen of A or B, but always as a citizen of A and B.
In exceptional circumstances and when justified, the Schengen countries can issue a return visa to its dual citizens, such that they can travel using the third country passport alone.
Indirectly for residents at least there is, non-Schengen nationals resident in Schengen are required to carry a valid Schengen residence permit - which as a Schengen national you won't have or need.
Everybody including the Swiss. I had to get of the train in Liestal and wait at the police station because the Swiss guy I was travelling with had forgot his ID card. We had to wait for his wife to bring it from home to the station.
In follow up to this - my wife only has a US passport and myself and the children have Swiss & US passports. Can/should my wife join us in the Swiss passport queue so we are seen together? Thanks.