An airport is legally within its rights to devolve responsibility of marketing to the public entirely to the airlines that fly there. However this is not always a good business model - an airline acts in its own interest rather than in the interest of the airport. As an example, Easyjet in London advertise that people should book a flight to go on holiday - in newspaper ads they do not specifically advertise (for example) Gatwick in favour of Stansted.
Even a heavily dominant airport like Heathrow spends money on advertising to the public in London. If it wants to boost passenger numbers generally, there is very good reason for Bournemouth to advertise locally in Dorset / Hampshire as well.