Take and bring in the sense of translocation do not have an exact, complementary usage bound by the location of the speaker as proposed by the question. Oxford Dictionaries defines this sense of bring simply as “Take or go with (someone or something) to a place”. Merriam-Webster defines the location binding of take as “to another place”, whereas bring is bound “toward the place from which the action is being regarded”.
The location binding of bring is not necessarily defined relative to where the speaker is currently situated. For example, in a telephone conversation, since the speaker and the hearer are not in the same location, to bring could be to the speaker's location, or it could be to another location contextually relevant to the conversation—“the place from which the action is being regarded”. You can say “bring your books to school” whether you are at school or at home, because you don’t have to actually be at school to regard an action from there. In context, you are simply imagining the action happening from the perspective of school.
Others agree. John Lawler parallels come and go with bring and take:
To summarize, both come and go mean to move, but their use is determined by their deixis, i.e, the identity and location of the speaker and addressee...
For instance, in a situation where someone has knocked on your door and you shout reassurance to them to let them know you're on your way to the door from somewhere else,... what you say is I'm coming, because you're moving toward the place your addressee is at; in English you can take either the speaker's or the addressee's position as the terminus ad quem for come, as well as the terminus a quo for go.
It's easy to see that bring and take have these stigmata, too.
I'll bring it right back. (to you)
I'll take it away. (from you)
Take this away. (from me)
Bring the car. (to me)
With this kind of fluidity..., there are lots of choices available for bring and take. If you are speaking to someone outside your office community, who will not be accompanying you tomorrow, you would be more likely to say I'll take the sausage to work tomorrow; but you could still say I'll bring it to work, because, after all, you'll be there, and it'll count as moving towards you, the speaker.
The Grammarist notes about hypothetical situations
When one is using the future tense, either of these verbs are correct because nothing has actually happened yet. Usage is based on which point of view the speaker wants to emphasize, the moving of the object or the removing of it.
The Shorter Dictionary of Catch Phrases (1994) defines it:
I'll see you in hell first a vehement refusal or a response to a challenge. The phrase dates from the late 19th century or earlier. Variants include I'll see you damned (or hanged) first.
This can be interpreted that it's more likely that the other person will be in hell before their claim ever becomes true. You don't necessarily need to go to hell as well.
Alternatively, it could mean you think their claim is so outlandish that it's more likely both will go to hell than it coming true.
The OED dates it to 1715, from Proceedings of the Old Bailey:
Saying G—d D—n him, twenty times over, and the High Constable too; he should see them all in Hell.
Another early quotation from 1879:
I'll see you in hell before I vote for Charlie Lake, or any other Democrat.
Finally, it can also be an expression of hatred, as demonstrated by the last quote in the OED, from a 2007 Independent on Sunday:
‘See You in Hell’ he sneered to two fellow death-row inmates he couldn't stand.
Best Answer
I'm not an English major, but I am a native speaker.
"I'll bring you home" is definitely not invalid; it's a perfectly fine thing to say, and I think your meaning is correct.
However, "I'll take you home" does not imply that you live at the same place, or that you're going to be staying over. I think it just implies a sort-of dominance on the role of the speaker. I would imagine this being said by a person speaking to someone who is more drunk than they are, or by a speaker who knows the way home better than the other person. Although, to be fair, it probably depends a lot more on who says it, how they say it, and exactly how they phrase it and not so much on bring versus take. For example, "I can take you to your place" has essentially the same meaning as "I'll bring you home."
I think the most natural thing to say in the case that you are both going back to the same place, or both heading home is "Let's go home."