Scenarios:
Player: "I think he's lying."
DM: "Roll Insight."
Player: "Um...4."
DM: "He rolls Deception 12. So you believe him."
Player: "Do you mean I can't tell if he's lying, or that I actually believe him?"
DM: "You believe him; his Deception was higher than your Insight."
DM: "The guard rolls Intimidate and gets an 18. Yeah, you're intimidated."
Player: "Oh, ok I guess I'll just keep moving then."
Player: "I don't think I want to do this quest for only 100 gold."
DM: "The magistrate rolls Persuade and gets...a natural 20!"
Player: "Darn. Looks like I'll take the quest."
Any reactions? How many people play the way that's described in those three scenarios?
Let me answer as both a DM and a Player.
Scenario 1: The obvious way to phrase this is not "that you believe him" but "you have no reason to NOT believe him". He has no tells, no ques, no outward displays of insincerity that you can determine. That said, there are lots of people I don't believe even though they give me no reason to disbelieve them either. As a player, I want the option to say "There's something funny about that guy, but I can't put my finger on it", and as a DM, I reserve a similar right to say "Ok, you might have rolled a 22 on deception, but nobody fully believes you're really a gold dragon polymorphed into a halfling".
Scenario 2: This is trickier. Rather than say "you are intimidated" I'd play up the "he's threatening you looks like he can back up those threats". In essence, the result isn't to see how the PC reacts, but how well the performance does: a guy who rolls an 18 keeps a subtle menace that shows he's capable of making life difficult (and do you want to bother with that hassle?) vs a 4 which says "this guy is blowing smoke, he's not really all that threatening, but he may yet be dangerous". The roll then measures the intimidators act, not the recipients reaction. As a player, if he tried to tell me I couldn't act or was fearful/cowed because of it, we'd have a major problem.
Scenario 3: Never never never would a skill check force a PC to accept (or deny) something they want to do. At best, this becomes a role-playing tool; it might influence their action, but never dictate it. The magistrate might be sincere in his beliefs, but ultimately the PC must decide to accept or deny. As a player, I might consider his words more thoroughly, but in the end I'm still deciding if I want to do it or not.
The big thing here is (and this based on my group, who are good sports on these things usually) the check never determines action, but it can influence it. An intimidated PC can still pick a fight with the guard or get into a verbal spat with him, but he is fully aware this guy is going to be trouble before he does this. The deceived PC sill can decide not to trust the guy, even if there is no outward sign of deception. The persuaded PC might stop to consider the magistrates words, but its his call to accept or decline the offer. Of course, the reverse it true for me as well; I can decide an intimidated monster might not give up or be bullied, a deceived NPC might not fully buy a PCs outlandish story, or a persuaded noble isn't going to empty his entire coffers on a sob-story.