I normally allow players to make their own rolls unless the outcome is something that may be hidden from them… which is fairly rare. As the Dungeon Master I will always hide my rolls, this way they don't which rolls I consider critical. I feel that if I don't hide some and then hide others it cues them that this is the time to really pay attention or to do something extra that they may not have considered doing otherwise. I want them to role play and make it as realistic as possible regarding what the characters may or may not know. In the RW we don't know if there is a sow bear with her cubs just over the hill so we don't know when to pay special attention or not.
The question of trust is indeed the principle concern I have in rolling for characters.
I'd like to think that my players trust me, but inevitably, some of those rolls made behind the screen will hamper the players in significant ways.
It gets particularly complex when someone has something like 5E's luck feat, which allows a limited number of die rolls to be rerolled.
I believe in narration and story over rolls of the dice. If the Dungeon Master wants to tweak the result for the story purpose I'm fine with that. This game is all about telling a story. In the current game that Aeric is running what if he had plans all along for one or more of our characters to fall into the Iliosani's hands. Carrick and Zinna had to make constitution checks to save vs poison and being captured. If Aeric wants to roll those and say one or both of us failed them I'm fine. It's part of the story. I'm not fine if that is done and your character dies due to the Dungeon Master fudging something. That is a deal breaker. I have no problem with the Dungeon Master rolling for me on those type of saves etc. If I thought my Dungeon Master was out to kill characters off then off course I wouldn't trust them to roll for me. Then again, I would be out of that game anyways.
Certain skills in 3.5 and Pathfinder specify that they should be rolled in secret, but it doesn't help the suspense if the Dungeon Master asks you for your skill modifier beforehand. Personally, I trust my group to avoid metagaming and let them roll for themselves. Sometimes the Dungeon Master misses a circumstantial bonus which the player wouldn't have forgotten.
The play by post format can be a little slow, so I often roll critical confirmation rolls, reactive perception checks and sometimes saves, but I will try to keep KNtoran's preference in mind for the future.