I think (...thought...) you know better. There's no invite process. You host the arena and if people join you see them on the roster.
They join the private arenas area and see the available options. They look at the rooms and choose one to join. They join it.
No PMs, no invites. This isn't facebook. If you host it, they will come. (aaaaand if you want to coordinate ahead of time with specific parties, i.e. duels and whatnot, just do that on private forums, or in PMs...)