How can you see what items the enemy demigod has?

No, there isn't a way to see your enemy's items as far as I know. Being able to do so would detract from the element of suprise you have when buying a new item. An enemy will probably run if he can see that you have mageslayer or ashkandor.

You can't.  You can't even see the items on your own people.  You can't even see mana on heroes either.

A very bad choice in my opinion.  As I saw in an interview it's an interface limitation and they didn't care enough either way to do the work to add the ability.

I hope that opinion will change in the future, as seeing enemy items was an important function in Dota that didn't break anything.

As do I.  Considering how big a difference they make, I think an opponent should have the opportunity to decide if they want to get murdered by Ashkandor or not. 

Or whether they have the heart of life or not.  So you can prevent them from using it.

Another thing is you can't even see what buffs or debuffs enemy heroes have on, aside from the ones with visual cues like Bramble Shield.

You guys have a point I guess. Although once your enemy suddenly becomes capable of killing you in 5 seconds without taking any damage you should be able to guess that they have some sort of artifact without having to see their equipped items.

Well sure, in Ashkandor's case but that insight comes a little late once you've already been destroyed by it.  

And then, of course, the other changes like HoL or just heavily stacked armor and such are a bit more subtle but still make a huge difference in the outcome of a fight. 

Another benefit would (in theory) be a decrease in some of the more ridiculous claims of imbalance.  Demigod X isn't overpowered.  You can clearly see that it's Demigod X equipped with 3 artifacts that's handing you your ass.

At a minimum you should be able to see your teammate's items.

I'd be perfectly happy with a history of purchased/sold items at the end of the match.

Knowing which one makes a huge difference.  Being able to calculate how much % dodge they have, for instance, is huge on knowing your chances when 1v1ing them.

I'd like to see what my allies have, but I think it would be a bad idea seeing enemy items. I've lured a few enemies to their deaths as they try to chase me and I pop heal as my ally flanked them. Wouldn't be possible if they could see the potion all the time.

I can definitely see how the element of surprise is good, but it sure would be helpful to anyone trying to learn the game.  I like the idea of seeing stuff like that after the game or as part of a summary/replay.  I think it would be extremely valuable to see how others played their DG, along with the items they chose.

Seeing what they have at the end of the game is not nearly as helpful as seeing WHEN they get them.

Possibly, but I think the benefits outweigh the loss of stuff like this.