Visa and Mastercard are umbrella organizations that at bottom work through banks; the card doesn't just say Visa or MC, it also has a bank on it, and the front part of the card number is a bank identifier. So, there's no reason that European banks can't also be part of the same network, and presumably they are.

European banks definitely issue Visa and Mastercard cards (and in many countries, there isn't even any local alternative for card payments).

But given the network effect, Visa and Mastercard have a lot of power over both issuing and acquiring banks. Issuers have at least a choice out of these two; acquirers just have to accept whatever their customer puts on the counter at payment time (or lose ~50% of their card-paying customers).

In other words, participation is (relatively) easy; getting some amount of control, let alone autonomy, is almost impossible – again due to the network effect.