You don't need privs if it's your own personal city. You request the construction permission for the faction as normal, and then you go down to Build Permissions and look at [Facility Requests (Player)].
"May the Grace of Ara go with you, and His Vengeance be wrought upon your enemies."
Are you trying to be owner of the facility as you go to build, or will the faction own it as construction starts? I believe if it's a faction specific facility, then the faction will have to own it at the start. Later you can ask for them to pass ownership to you.
Also, some larger types of facility cannot be build on certain types of terrain (e.g. Mountains/Craters/Glaciers: nothing bigger than 5x5.)
There is no god but Ara... All will walk his path... The Faithful of Ara <--- (Warning: music link)