People never thought to ask Gai about his rivalry with Kakashi. Kakashi fielded more than one question about their rivalry, but no one asked Gai. Gai knew that Kakashi probably dealt with the questions with a mixture of his usual bluntness and evasive charm, revealing much less than he covered up. That was Kakashi's way of going about things, and Gai didn't mind. But he knew no one would come away with an accurate picture of their relationship by asking Kakashi.
On Gai's part, he didn't feel there was any need to exercise secrecy. But people still didn't ask him.
So he watched people walk around with the wrong idea firmly fixed in their heads. He didn't mind. He felt no need to volunteer the truth. Still, he worried that one day Kakashi would get out of hand with the fake explanations, and they would end up with genin teams that had entirely the wrong idea about them, and that would necessitate him stepping in and clearing up the controversy forever.
Part of him liked the controversy. So he didn't really want to clear things up if they had to. For one thing, the assumption that he was Kakashi's sex partner was very flattering. It also amused Kakashi, which could only be a good thing. Few things afforded Kakashi genuine amusement, and the idea that they locked lips – and other things – on a regular basis was one of those things.
Still, if anyone had asked Gai to begin with, everyone in Konoha would know they were friends.