This fight can go to either team, however, racials (specifically Undead and Dwarf) give huge advantages in this fight, so hopefully you don't fight a double Undead team since that, in my opinion, is the hardest. This fight can go the following 3 ways:

Your Priest on their Priest and their Rogue is trying to CC your Rogue:
Follow your Rogue around (Priest will try to bring him behind a pillar to Fear him so you can't dispel), avoid Fear from their Priest and try to get as many Mana Burns as possible on their Priest, spam Dispel on his HoTs so your Rogue does more damage. If this is the scenario you should come out the victor, but be aware of a Fear from the Priest, Dispel your Rogue if he does get Feared and try to Fear bomb both the Priest and Rogue to force the Rogue to use his trinket then you can Blind him.

Rogue vs Rogue, Priest vs Priest:
My favorite style, Dispel the Rogue of his HoTs and Shield, be aggressive on the Priest but DO NOT GET FEARED. Try to time your Rogue to Gouge the Priest so you can run over and Fear him without being in danger. If your opponent is Undead and you are not, be very conservative, he realizes he has the advantage over you and will go for you. You should save Fear Ward for when you run up to the Priest (he will Fear you when you are in range). Cast Fear Ward right before you are in range and there is a chance he will blow his Fear and waste it. If he does, immediately Fear and Mana Burn him, if he is Undead back off and Mana Burn (Shield yourself so the damage from the Mana Burn doesn't increase your cast time).

Use Shadowfiend on the Priest and Dispel the Priest if he Shields (if he Shadowfiends you, run behind a pillar and then Shield yourself. If the Priest tries to come Dispel you get ready to Fear him). This fight heavily depends on which Priest goes OOM first and which Rogue does more damage to the other one to make the Priest heal more and be more open to Mana Burns. Obviously whatever I say in this guide the other Priest will know as well, so create your own strategies and tricks on how to counter this scenario.

Race to kill the priest:
Dispel the Priest whenever you can. Fear the Priest then go to kite the Rogue around a pillar, if he stealths, run away from the pillar but not around it since he will catch up to you. Do not get Shadow Sight, this now makes you take 5% more damage and even though you can see the Rogue he eventually will get back on you by mounting up or sprinting, so don't bother. If the Priest is in your sight, spam Dispel and Mana Burn to make the Rogue blow his Kick on your Shadow school.

This fight depends on which Priest can dispel the Rogue of his Fear better and which Rogue does more damage. If the Rogue trinkets after your Fear (if the Rogue who is on you isn't Undead give yourself a cheer), have your Rogue Gouge the Priest (so he doesn't run far away). Make sure a new Crippling Poison is applied, then go Blind the rogue. If your Rogue is in this same situation, and you see the other Rogue coming to you, use your Evasion and maybe you will Dodge it. Their Priest will get free Mana Burns on your Priest and the Rogue will do a ton of damage, so do not trinket Fear in my opinion, ever.

Be sure to switch targets to help your Priest and coordinate yourselves. Mirror matches of this type are really fun, the more creative and coordinated/lucky(stun resist and the likes) team will win.