I seem to be a mix between Stoney and Chronovore. I enjoy sandbox games when they give you a story, but I mostly only do so at the most basic of levels. This is why I love GTA3, Vice City, Saints Row 1 and 2 so much, because you get to play a specific character (gangster) with a specific goal (be the best gangster) in a specific environment (gang ran city). It gives you something to strive for. They are games that put you in a role and it's your job to build on that. I don't really require much narrative in these games because the actions mean that much more. Seeing Niko complain about his past and then turning to murder even more doesn't move me; I don't give a shit about his pathos. I just wanna have some fun with a fun character.
It's a part of why I like role playing games so much: you get to play as someone else, someone usually far more interesting than me and that's why I tend to prefer playing as characters who have no moral compass or completely different goals than me. I don't like playing as the good guy in these games, because playing the good guy takes you out of the whole "hey, I'm running over and shooting people" mindset. It ruins the setting and breaks all immersion for me.
On the other hand though, most of my time spent playing sandbox games is doing crazy stuff like stunt jumps or odd jobs. Things that really test me as a gamer, both in skill and patience and my "I want to throw my controller" meter. I really get a kick out of exploring a new GTA city from head to toe just finding little things to do, which is partly why I found GTA4 so disappointing and yet RDR so riveting.
The best sandbox games to me, are neither one side or the other, but a mixture of both (GTA3, Vice City, Saints Row 1/2, RDR).
Basically, if your character's resolve doesn't gel with the shit you're doing, the sandbox game feels like a flawed experiment, because the most important aspect of open-world games, above all else, much like rpgs, is creating a believable world that the player wants to soak in and absorb just because it's fun and interesting to do so. But it's equally as important to provide a number of activities to do, because half the fun of these games is exploring a made up city for you to toy around with. The tricky part is providing activities that make sense to the setting, characters and environment. Otherwise, you break the rule of immersion.