How should membership of the various continent-spanning guilds be handled? Are there going to be 3 competing Fighters Guild, Assassins Guild, etc?

Arcane Guilds - College at Winterhold (Ebonheart), Psijiic Order or Thalmor (Aldmeri), Mages Guild (Daggerfall).
Martial Guilds - Companions (Ebonheart), Syffim? (Aldmeri), Fighters Guild (Daggerfall)
Assassins Guilds - Morag Tong, Dark Brotherhood, ?


Or should guilds span the continent and perhaps take a neutral position regarding the conflict? If two members of the same guild in opposing political territories happen to cross paths, how should that relationship be handled? Which faction takes precedent, the guild or the state?

Guilds should be neutral I guess.

I think perhaps each faction should have a version of the major guilds, although not pure mirror images of each other, but also should have their own exclusive faction.

Well, since the Fighters and Mages guild are supported by the empire... and there is no empire... all is possible

Well there are some factions that are very similar that are within the same factions. Like House Redoran and the Companions being the Fighter guild variants. Idk if you really can make just three guilds for each faction that are similar to the Thieves Guild, Mages Guild and Fighters Guild.

If I had to give the Daggerfall Covenant an Assassin faction it would be either the Alik'r or the High Rock spies. Dunno if they have a given name but they are supposed to be some of the best spies in Tamriel.

They aren't including the college at lauch, so we can rule out that concern. The Psijiics only allow the best-elite devoted mages, perhaps the PC isn't quite good enough. The mages guild is about teaching, so they can easily join that. Hence, there's little conflict there.

The other two? They may be faction specific, although one of the devs is talking about earning rep with the FIGHTERS GUILD, so perhaps they haven't considered the others.

All in all, I imagine your second statement is accurate. The guilds may take a backseat in the conflict. Alternitabley, faction specific guilds could exists, whose alligences are to-well, those factions.