No private entity has the right to conduct war.

Yes, but under international law that nice thick line starts to show some wear and tear.

Mercenaries are not accorded the rights of lawful combatants in the Geneva convention, however so long as private military personnel are:

1. Nominally acting under the authority of a state authority.

2. Nationals of a state that is party to the conflict.

Thus, Blackwater in Iraq was legally an agent of the US government, so that where it employed Ukranians as employees they were lawful combatants (because Ukraine was a party to the conflict), but when they brought in former South African special forces they employed unlawful combatants, because South Africa wasn't party to the conflict.

Murky, eh?

And what happens when the size of the these PMCs armed contingents is larger than the local army?

And I'll give my consent to any government that does not deny a man a living wage-Billy Bragg