A friend and I want to venture into Daggerfall/Ebonheart territory, and ideally be able to do quests there. Is this possible?
While Bunyip's answer is basically correct, it isn't 100% complete:
You can't enter other alliances' territory through Cyrodiil.
The gates leading to their provinces (High Rock, Morrowind, and Elsweyr) are considered safe zones where players can spawn, organize etc. They're walled off (with one way doors back only usable by the owning alliance) and heavily guarded as well.
Once you've reached level 50 and completed your main quest storyline as well as your alliance's storyline, you'll be presented with the option to play one of the other remaining alliances as veteran content (often referred to as 50+).
This will allow you to play all of that chosen alliance's content with your existing character while having everything upscaled to level 50 (so it's still challenging and you're not just breezing through heaps of low level enemies).
Once you've finished your second alliance playthrough as well, you'll be presented with the option to play the last remaining alliance's content as 50++ which will once again allow you to enjoy the content with raised level/difficulty to accomodate your more experienced character.
As a side note: I haven't heard of any rumors changing this, simply because of the fact that it wouldn't fit well with the storylines (and the content is accessible already). Take the Ebonheart Pact storyline as an example. Without spoiling anything significant, it starts off with an invasion of Daggerfall Covenant forces in Skyrim and Morrowind. Why would they allow any member of the Daggerfall Covenant to join their ranks driving them back? This doesn't make any sense.
Well apparently you can now. I know this reply is old... but I'm Daggerfall Covenant and accidentally found my way into an obvious Ebonpact land. Strangely, the Daggerfall soldiers in that land are all red to me and will attack me now so I guess there is not really any faction in the usual meaning of the term.