This sounds very much like The Narcissus Plague by Lisa Goldstein, first published in Asimov's in July 1994.
As in the question, a virulent disease causes people to endless to talk about themselves, just spouting a never-ending rambling flow of drivel. The story is centered on a reporter, Amy Nunes and her editor, who take extreme hygienic precautions to make sure they do not catch it. One way to check each other is to ask after each other's health:
"Hi, Amy, how are you?" my editor Thomas asked. This is the only way
we greet each other now. It's meant to assure other people that we can
still take an interest in them, that we don't have the plague.
A co-worker, who had always thought himself the center of attention, suddenly catches the disease and becomes much, much worse:
"I like to be noticed," Gary was saying. "I love it when people pay
attention to me. That's what I live for. I have to have someone
listening to me and watching me at every minute ..."
Almost everyone was trying not to laugh. "One day, I remember, we
were sitting around and talking about the president," Gary was saying.
"So I started talking about the president too, and then the
president's brother, and then my own brother, and finally I got to my
favorite topic, myself...
One of the more enterprising reporters on the paper had turned on his
tape recorder. If Dr. Clark had indeed found a cure for the virus Gary
was going to have a very hard time living this one down.
Amy goes to interview a scientist, the Dr Clark mentioned above, who claims to have found a cure for the disease. It all seems to be true, and the doctor keeps a small supply of the pills in a bottle on her desk. She cannot release the cure until the FDA gives approval, so Amy schemes to steal the bottle and dose her boyfriend, Mark, who has been unbearable since he caught the disease. The story ends on a happy note when shortly after taking the pills, Mark's monologue tails off and he asks her "Where have you been?".
