Personally I won't have the staff for another month at the earliest, but there are warlocks who already have it, and I found a WoL log featuring one of those: World of Logs - Real Time Raid Analysis
The above link crops the time interval to the portion of the raid during which he had the staff equipped. One of the simplest ways (and really the only practical way outside of controlled testing) to get an estimate of the proc rate is to count dot ticks and Wrath of Tarecgosa ticks: Over the full BoT clear in the log this works out to 1298 / 8697 = 14.92%.
It's immediately apparent that Wrath of Tarecgosa actually double dips from damage taken debuffs on the target, so dot heavy classes have an inherent advantage - in a normal raid environment WoT causes 108% rather than 100% of the damage of dot ticks that proc it. Of course this has hilarious effect on a target like Halfus:
[23:20:09.372] Dangles Bane of Doom Halfus Wyrmbreaker *241319*
[23:20:09.939] Dangles Wrath of Tarecgosa Halfus Wyrmbreaker 1042498
There's not much more interesting info to glean from this single log. Affliction doesn't have a lot of direct damage spells, and the few it does have are pretty straightforward. Procced Haunts and Shadow Bolts refresh/update their relevant debuffs on the target, but that's pretty unimportant since those debuffs should be up 100% of the time anyway.
So I'd like to invite any other locks who have the staff to post their logs here, either from raids or from controlled testing on dummies. Of particular interest is logs from the other two specs, to see if the proc rate is different. I'd also like to see if immolate can proc a copy, and what exactly happens when it does - I don't expect it to spawn a second immolate dot on the target, but it might count as a refresh, resulting in an extra tick at the end.