The video games don't offer much with her personality, so I'll address the anime and Special.
Anime: Of all the Kanto GLs, it's astonishing Sabrina's given all this detailed back story. Certainly one of pathos and quite the daddy issues (since the powers are from his bloodlines and he pushed the training at the extent of all others), and the doll angle is refreshing. Although, in terms of archetypal personalities, she has all the cliches: quiet, borderline dual personality (even creating a Scarface/Ventirlquist setup, Batman fans will know what I mean), stoic to the point of being a Vulcan. And unfortunately, her final episode doesn't offer much in terms of personality, beyond the telepathic connection to her Kadabra. It's a mixed blessing (as one could say regarding the most emotional stunted Kanto GL, first only to Giovanni perhaps).
On another plus side, Lisa Ortiz offers quite the voice with her, and that should be commended.
Whereas the anime one has its shortcomings, the Special one succeeds.
Special: Of the Kanto GLs, she is unique as she has proven she has more dimension, given the series' standard treatment of her.
One: Unlike the rest of the Elite Triad, she was the only enthusiaistic Rocket vis-a-vis wearing the uniform, as opposed to Koga, who did initially and then discarded it. Then there was Surge, who was more neutral, but later cared, as seen in the Silph showdown and the G/S/C saga. Either way, Sabrina was the true believer of the bunch.
Two: Indulgence. One major theme with Sabrina is indulgence and the ability to outlet herself. Since her doll was absent, the writers offered a personality shift whenever she was in Battle vs. her in conversation. Her emotions surge out most in the game, there. Special, while not addressing it like the anime (like in the laughing scene with the Haunter), plays with her subtext given in Three.
Three: Humanity. This isn't an issue of compassion, but more on herself as a person and her interactions with people. One being her female side as shown with Green's breast joke and her insinuations toward Giovanni (creating an interesting ship) in the Silph showdown. Other examples would be her acidic relationship with Koga, and the amused/bemused interactions with Green in both the dubbed arcs.
She comes off as a person, not a caricature. On the down side, Special offers no back story or any reason to care about her, beyond what is laid before you. It's a shame, it really is, because it would make for such a delicious package.
That is my take, for what it's worth.
I'll correct the spelling later.