Re travel restrictions within the US

Following an item on BBC radio this morning, I had a look at the CDC recommendations for travel with the USA.It says that if you are fully vaccinated you can travel within the US and do not have to self quarantine on arrival. It does also state you should follow local recommendations too.

How does this work in practice. Do airlines, hotels etc ask for proof?

I ask the question because here in the UK there is talk of maybe people having 'vaccine passports' in the near future.