I'm not sure what people will think if it goes the way of Smeagol/Gollum, but I'm sure it won't go that way. Hm. What do you guys think will happen once that aspect is realized?
Well, if I'm interpreting the quote/double quote divide correctly, then from the perspective of everyone else Azula is already talking to herself, and not in the sort of musing way that many perfectly normal people do- she's addressing herself as another individual and using "we". For all that she's generally the only one in the room when she does, that's not a habit which can be concealed in the long term- particularly in a setting where people are explicitly superhuman in a way that at least to some extent includes sensory abilities like hearing. Switching who is in the driver's seat in the middle of conversation, as she's apparently now doing habitually as well, is also likely to be picked up on by others, particularly highly skilled others (like, say, Lo and Li) who will notice the shift in body language, attitude, and tone.
So one thing that they have incoming is people- at least some, possibly many depending upon how/if rumors are spread- coming to believe that Azula has a split personality, which makes her either possessed or crazy depending upon one's perspective. How they react to this will almost inevitably strongly affect the relationship between the two of you, because if, say, Ozai decides that a split personality is an imperfection that must be excised... well, that's a whole plotline in the making. And the cadre that Azula's building- they'll be closer to her than anyone else, so they'll inevitably notice and gossip with one another about it if nothing else; sooner or later (probably sooner) there will be a prevalent opinion amongst her minions about what the fact that Azula believes herself to be two separate people means, and Azula will notice.
The whole "I am you" concept was a very strong opener which could hold up almost indefinitely under casual scrutiny, and indeed from a certain perspective is actually the case. However, if placed in an environment where there's constant pressure on Azula to become hostile to, ignore, or attempt to excise her friendly other self, relations could deteriorate rapidly. The fact that body control is shared means that things could turn extremely ugly very quickly unless carefully managed, but on the other hand Azula's sources of positive reinforcement at the moment total exactly one, and it's the voice in her head that is with her 24/7, which is a very strong position from which to carry out emotional manipulation. Still, I suspect she'd utterly loathe having most people consider her critically flawed, and there is a very strong risk of that becoming a widespread reputation if things go as they currently are.
As for "not realizing it's a dream" and freaking out, at this point Ms. Headvoice absolutely cannot afford to freak out and should know it. Anything which reveals to Azula that the initial identity claim isn't true will make for an immeasurably less pleasant living environment in the however-long-this-lasts to come, and that's potentially an entire lifetime. A private bit of gibbering or nervousness could be okay, but anything that Azula can hear, or which involves taking over the body for a while? It simply can't be countenanced. Denial/repression of all existential panic issues, possibly forever but at minimum until Azula is so emotionally close to her head-twin that she would forgive her anything, is mandatory to avoid horrible, horrible things. Tiger riding, go!
Anyway, from previous works I'd say that the ability of this character to accept and roll with things is remarkably high; sharing the body of a Fire Nation princess isn't going to do more than slow her down a little as long as she doesn't get tied up in dwelling on dark thoughts, which she won't because she's busy effectively raising a child into a ruthlessly effective warlord and plotting world domination, and those are time-consuming matters. The first serious emotional shattering won't occur until after she's inflicted huge amounts of suffering on others while attempting to optimize her course of action, at which point she'll lose it briefly, probably have some kind of relationship-redefining moment with Azula, and then proceed onward to inflict further suffering.