Now this is fun. Shadow priests have the lowest survivability among the three specs but compensate with a lot more offensive punch. While shadow priests aren't a common sight in upper bracket Arenas, they thrive in the Battlegrounds so expect to see your fair share of priests in Shadowform. Because of that key talent, shadow priests are easy to spot. The caveat is that most shadow priests in the Battlegrounds know how to PvP in the same way that a holy priest is likely to be a PvP novice.
More important than offensive dispels when fighting against a shadow priest are defensive dispels or the ability to remove their debuffs. Shadow Word: Pain is standard fare and there are a ton of talents in the shadow tree to boost its damage to obscene levels. It is also in the priest's best interest to keep the debuff on you because of its synergy with Mind Flay and Mind Blast. With the talent Vampiric Embrace, all of a shadow priest's single target offensive spells turn into heals, as well, making up for their losing access to holy spells. Essentially, all their spells act like Devouring Plagues. Speaking of which, Devouring Plagues also have an instant damage component when used by a shadow priest, making it pack a punch even when its dispelled.
If you can remove debuffs (one is a magic effect and the other is a disease), keep them off as much as possible, even it turns into a war of mana attrition. Neither debuff is cheap and will cost the shadow priest in the long run. Shadow priests also have the AoE toy Mind Sear and can wreak havoc in the chaos of a melee engagement, but this leaves them vulnerable as with their other long cast or channeled spells.
A shadow priest works similar to a warlock in that they rely on the fear effect of Psychic Scream, stacking DoTs, and casting damaging spells while you're crowd controlled. They can use it a little more often and will do so at every opportunity. As mentioned in the first part of this guide, Psychic Scream is worth the cc-break. If you have spell interrupts, try to let Psychic Scream take effect for a bit and break it only when the priest is channeling Mind Flay or is about to cast Mind Blast. Otherwise, break it immediately to prevent yourself from being feared too far away from the priest. Keep track of Psychic Scream with AddOns such as Afflicted 3. Know when they use it, and stick to them like glue when its on cooldown.
Shadow priests play hit and run perhaps more than other caster classes by applying Shadow Word: Pain, running, casting Mind Flay to tick for a few, then running again. When they've created some distance, priests will try to keep Mind Flay active to ensnare opponents, as well. This is a magic effect and the channeling can be broken. Against healing classes, shadow priests will try to land a few Mind Blasts to keep a weaker Mortal Strike effect on the target.
Keep the distance close. Melee range will dissuade the priest from casting channeled or long cast spells. This is easier said than done, as shadow priests have a solution to snares with Improved Shadowform, which turns their Fade into a snare removal. PvP priests will also have Fade glyphed and talented, allowing them to break free of snares and roots every 15 seconds. This makes them harder to pin down, so keep track of when they use Fade and time your snares to land while its on cooldown.
They also have other tools at their disposal, such as Silence, which they'll often use as a precursor to casting long cast spells such as Mind Blast. When they use cast Silence, try to close the distance as much as possible and bother them with melee attacks. Because it's a blanket silence, most shadow priests don't time it to coincide with spells as a counter and use it freely. When they use it, it usually indicates that they're about to go on the offensive, so be prepared for it.
On the defensive end, shadow priests can use Psychic Horror as a peel against bothersome melee. Be prepared for this. There won't be a need to use a trinket against this as it only lasts for three seconds, but the main component of the spell is to take away your offensive punch. It has a long, two minute cooldown but saves shadow priests when timed right. It is also a horror effect, which means it doesn't share diminishing returns with Psychic Scream, so they can use both spells in succession to keep you out of play for a longer time.
Finally there's the often ridiculed Dispersion. Taken by all shadow priests as a necessity, it is a nonetheless lackluster last resort mitigation spell. Forcing a shadow priest to cast Dispersion is a good sign that the fight is going your way. Despite the huge mitigation, keep attacking and following the priest at this point because she will have few options left when the effect wears off. It's almost like when a paladin casts Divine Shield, priests only cast it when things aren't going well -- either they're low on health, mana, or both. The idea is to burst a shadow priest down, because they'll simply survive long, slow fights. High burst damage is problematic for shadow priests because they are unable to quickly recover from it the way healing priests do. Keep the damage output high and make sure to break their limited crowd control spells.
Personally, I think shadow priests are the most fun kind of priest to fight. They pose a different kind of challenge as opposed to merely trying to bring a healer down. The strategy to fight them is moderately different in that you'll need to keep yourself clear of debuffs and it will help to keep them without a Power Word: Shield. Movement will be important, too, because shadow priests will keep on running. Finally, shadow priests are the only priests who are experts at First Aid, and they'll bandage when able to save the mana from switching between forms. Have fun and good luck!