It's not really much of a wonder that the other diary users knew who 1st and 2nd were, because they're the ones who used their diaries so openly due to the fact that they're kids who tend to rely on them way too much. In fact, let's look at how the others knew Yuki was the First:
Third (the serial killer)- discovered it because Yuki used his diary to ace his math test (and it was Yuki and Yuno's defeat of him that caused the others to take an interest in Yuki)
Ninth (Minene)- same reason as Third, as Yuki's test results did spread outside the school
Fourth (that cop)- followed Ninth in order to discover this fact
Twelfth (that super hero masked guy)- actually, he never really knew this
Sixth (the priestess girl)- a combination of Fourth's fault and Ninth's fault
Fifth (the kid)- he found out during the incident with the Sixth
Tenth (Hinata's father)- pure coincidence, as he was chasing Akise at the time, and Yuki happened to spill the beans during that dog chase, causing Hinata and Mao to turn their attention towards him
Seventh (Ai and Marco)- Ai was undercover as a shrine maiden in Tsubaki's mansion and found out during that cult fiasco (watch Episode 5 again and you'll see her trying to treat the Sixth's hand when it got cut off)
Eighth (that big headed woman)- same reason as Seventh and Fifth, although she wasn't actually there. It was one of her apprentices (again, watch Episode 5 and you'll see that one of Eighth's apprentice holders was trying to treat Sixth's hand along with Ai)
Eleventh (the unknown one)- he's obviously a guy in a high enough position to monitor Japan effectively, so there are a number of ways he could have found out
And do I have to explain how Yuno knows?
Bottom-line, Yuki's bad luck and idiocy caused him to get exposed to all the diary holders, which is why he and Yuno are usually the ones fighting the other diary holders. Everyone else was more careful about exposing themselves, so the lack of free-for-alls is understandable. And no, I do not know why Ai didn't attack Yuki and Yuno after they defeated Sixth or anything like that, so don't ask me.
A story shouldn't just give people what they want, but also what they didn't know they wanted.