Dubuquedogtrainer gave you such a thorough answer for addressing the barking issue (plus a link to more info) that I won't try to improve on that.
I would, however, like to ask you if perhaps you resent the dog's "real owner" more than you resent the dog's barking, but are directing your resentment toward the dog because that seems preferable to a confrontation about something else.
You make a point of stating that it's not your choice to live with the chihuahua, but mention that the owner "...also lives with us." Perhaps this extra housemate is the relative or friend of the other part of "us"--your other housemate (spouse? partner?) , whom I presume you do want to live with, but you feel constrained to allow the third party's presence rather than upset your relationship with the second.
I realize I may be reading something into your question that isn't there, but if I am right, I hope you'll address the main problem for your own peace of mind and happiness. Anytime one feels forced to live with a person not of their own choosing, resentment may occur. If the main issue is not confronted directly (albeit tactfully), that resentment may take over your life and spoil your relationship with the "second" party.
If you weren't accustomed to being around a small barky dog before this one moved in, I'm sure it's probably getting on your nerves, and I hope Dubuquedogtrainer's suggestions are helpful in that respect. If my intuition was working properly when I read your full question, I also hope you'll take steps to address the underlying issue, even if it requires getting counseling to cope and be comfortable with a living arrangement you may not like.
If, on the other hand, I was dead wrong, please accept my apology and consider me "butting out."
Whatever the case, good luck!