Some additional details have been published in ComPet Bulletin #6:
"The two players take turns choosing what ability their pet will use next. The key to success in battle is to manage your pet’s health, mana, stamina, abilities and cooldowns in order to overcome your opponent. Just before the battle begins, you choose your ability set and apply buffs to your pet in order to maximize the effectiveness of your strategy and play style.
In the final product your pet will battle in vibrant environments, and have a vast amount of different abilities to choose from, which will cater to all play styles and strategies."