The old zones, like Thundering Steppes and up, do not have bases or the likes restricted for only one faction. Maj'Dul has three different factions, and you will only be on the good side of one of them, the other two will always try to kill you. (you start with all three hating you though, need to do quests and a lot of them) Ofcourse, this does not mean that a player from Freeport will be able to do all the same quests as a Qeynos player, not every NPC will talk to everyone.
Kingdom of Sky zones don't have a lot of faction zones or bases.
In Faydwer, there are cities, but only in one will a Freeport player get attacked.
In the RoK zones, all are equal again. Meaning hated at first.
And I recall doing quests in Freeport with my monk from Qeynos, more specifically for the Freeport Militia. For quite some time, my faction with the Freeport Militia was much better than my faction with the Qeynos guards, it still is better than the Royal Guards for some reason.