Raising the post from the dead for a comment.
Both of them are very prone to not trusting anybody with everything. It's an attempt to prevent a problem with putting all of your secrets into one basket, so to speak. If you do so and that one person gets compromised, well...
With regard to the whole refusal to discuss his family until Conan comes clean about the specifics of his poisoning and identity, I can see it as a sense of wanting some kind of reciprocity if he does end up explaining the details. Your dad getting murdered by a crime syndicate you are hunting after is kind of personal and not something you share with anybody unless necessary, even if they are trusted. Same goes with Kudo's poisoning. Shuichi ribs him once when he witnessed the voice changer, to show that he figured things out, but otherwise avoids bringing it up.
As for them both researching Rum/Asaca separately, I think that has more to do with FBI procedure. To my knowledge, the FBI try to avoid including civilians during investigations if they can help it.
"It is a capital mistake to theorize before one has data. Insensibly one begins to twist facts to suit theories, instead of theories to suit facts." -Sherlock Holmes