Traditionally, and this is still the case in many countries, no. Airlines used to just check requirements, and they didn’t transmit anything to anyone.
However, over the last couple of decades, this has changed a lot, and many counties require information from airlines either for incoming passengers or exiting passengers, or both.
This takes the form of API (advance passenger information), also known under a variety of different names in different places.
If an airline asks for your passport info before they let you check-in, it’s for the purpose of transferring it to either the destination or origin country.
- all passengers going to the US (or even flying over the US) will be notified to the US. They will be checked against no-fly lists, and for those going to the US, their eligibility will be checked (valid ESTA or visa...). If you’re not eligible, you won’t be allowed to board.
- all passengers leaving the US or the UK will be notified to the origin country. Neither has exit immigration checks, so that allows them to update records and notice overstays.
I believe the EU (or is it the Schengen area?) already has a similar system for incoming passengers. Other countries already have or plan to have the same.