OOC
There's a difference between wanting to keep character information private (e.g. having a patron, secret or what-have-you that might be considered antagonistic to other players'...for example, having a positive relationship with the Lich King when there's a known Cleric of The Priestess in the group) and actively being antagonistic or hostile as a player or character.
Keeping a character trait or relationship hidden can be an effective tool in building character dynamics between players while avoiding preconceptions, especially in a group of strangers who might jump to false conclusions. Yes there is benefit to being open about such things when stereotypes are being adhered to and/or there is an understanding that Player X can already be trusted not to abuse the trust of other Players when playing to less-than-Heroic ideals, but there's a marked difference between "hiding the truth" and "lying" in this context. The former does not preclude building trust. The latter does.
As a long time Rogue player, I've often had to toe the line of moral ambiguity and frequently faced prejudice and preconception about who or what my characters are. Very often I've had to correct other players (particularly on PbP) about such things as being a thief or criminal when the truth couldn't be further away. If my character is a Lawful Good noble duellist that just happens to be of the Rogue Class, I feel like I'm justified in not telling other players that I'm playing a Rogue, because all too often I've faced false accusations of theft, larceny or of being untrustworthy just because some stranger has decided that Rogue = Chaotic Neutral, despite all relevant evidence to the contrary. I've literally had a Paladin player attack my LG "Private Eye Ex-Cop Investigator" Rogue based solely on my being a Rogue and the GM telling the group that some of their gear went missing. Yes, that player was super toxic, but had they not known my Class they'd have had no reason to jump to the conclusion they did. Regardless of the circumstances/reasons, my open honesty in that case actually caused friction in the group and failed to build trust.
As I said before, I'm happy to be open about Icon rolls if that's the group/GM preference, but I thought I'd explain why I would prefer at least certain aspects to remain private, including our Icon relationships and Backgrounds. At least until they come into play and/or our characters have the opportunity to learn that information about one another organically in the narrative. If, at that time, it causes friction then at least it's in-character friction between the characters that can be amicably resolved out-of-character by the (hopefully mature and trust-earned) players.