Most (all?) of them were privatized a good while ago already, and there's been various mergers and bankruptcies between the national flag carriers, but the ones remaining indeed still tend to have a dominant presence especially on international flights (obviously for many European countries, domestic flights aren't a big business if they exist at all, which is one obvious reason why there was kind of a natural monopoly in each country for the state-owned lines). Nearly all of their new competitors of recent decades are competing in the low-cost market segment, and indeed mostly without flying far beyond Europe - at most to holiday destinations in North Africa. Virgin is an exception coming to mind, having done trans-Atlantic flights for a long time. Increasingly, the low-cost airlines are now entering that market as well.