Immigration officers from the country you are leaving usually do not care much. I guess some might do something if they notice a problem but checking whether you are likely to get in trouble later on is not their mission. Note that some countries don't have any formal immigration check on exit…
Airlines everywhere do check whether passengers have appropriate documentation for their journey but that's mostly to avoid being fined by the destination state, not to help travellers. So what airlines have to do is merely checking that the visa “looks” right, to the extent required to avoid liability if a passenger is refused entry afterwards. They do not and cannot check if your name is in one database or another or whether you will actually be granted entry.
Do note that there are countless combinations (hundreds of potential destinations, each with several types of visas and different sets of visa requirements) and it would be very complex to check them thoroughly, let alone ensure that countries share information with all other countries on the globe.