This daily mission is given by the Junior and regular Agency Handler from the office in Rome Blue as well as the office in Athen. It requires you to travel to the south west corner of Belial Forest and speak with the Dog Trainer.
Enigma Dog Trainer Location: 530 x 865, Forest of the Unborn, Belial Forest.
From what we have found so far, only one person appears to be able to speak with the Dog Trainer at a time. When you speak with him, you need to advise him that they said something about "training enigma dogs".
Enigma Dog Trainer
While speaking with the Dog Trainer, you should receive an item in your inventory
Note: You will NOT receive this item if you are a bureaucrat.
This item can only be used for mindcontrolling pet dogs. It also cannot be just any kind of dogs, it has to be Enigma Dogs. The Dog Trainer has quite a few dialog options to give you information about this.
Once you are ready to get started, you need to locate an Enigma Dog to train (charm).
After speaking with the Dog Trainer, you should have a marker on your compass showing that you need to run North. Occasionally you might find an Enigma Dog between the Dog Trainer and this marked location.. however more often than not it's faster to just go to the marked location. Very often you'll also find Enigma Dogs in the Whom-Pahs to the bars in the player city to the north. (The dog leaves its owner when the owner zones and thus the dog stays near the last location of the owner.) In order to calm one of them you must target them blindly (using the TAB key) while standing near those Whom-Pahs because they are behind the closed doors and you cannot see them.
When you find an Enigma Dog, target it and then right click the Charged Nano Critter (Temporary Charm Enigma Dog)
in your inventory. Your dog should now be loyal to you.
Training your Dog
To train your dog, you need to locate five of the ball constructs in the area. You then need to command your Enigma Dog to destroy them. You can do this using /pet attack
Each ball that your dog destroys will update your mission and tell you how many more still need to be killed.
It is a good idea to pay attention to the Swamp Wargs in the area. These mobs will be agroed by your pet dog and you will need to kill them yourself.
When your dog has destroyed all five of the required ball constructs, your mission will update advising you to return to the agency.
It would be very nice of you if you would release your dog with /pet free and kill it yourself or let it be killed by the mobs in the vicinity before you leave the area. The dog will then respawn in the mission area and other people will find it more easily. Professions who can use the Grid to get back to the main cities can do so. The pet will stay behind. You can also use /pet wait to leave the pet near its current position. (Credits to Lliers and Gatester who pointed this out.)
Note: As an additional reward to this questline, before returning to the agency, you can return to the Dog Trainer. If you tell him that you had a lot of fun training the dog, he will offer you an Enigma Dog Pet Nano for 30k credits.