The eight-week-old Labrador and 10-week-old Malinois have been bred in Scotland specifically for police work and are already showing skills.
PC Gareth Wedley, who is Luna's handler, said the Labrador was chasing and fetching a ball earlier than any puppy he had seen at her age before.
The pair have joined the 17 other dogs in the force's dog section.
Luna the Labrador - named after a character in the Harry Potter books - is living with PC Wedley and his other working dog, a two-year-old general purpose dog called Axe.
"Luna is doing really well at the moment - she's already displaying a desire to chase and find a ball and bring it back to me, when she wants to give it back that is," he said.
"This is something which I haven't seen in a puppy I've had of her age before.
"She's currently undergoing some training and we are also trying to introduce her to various environments.
"Early socialisation and introducing the dog to a wide range of people, animals and experiences is essential to ensure a sociable, well-adjusted and balanced dog which is suitable for police work."
Diesel, who has been living with handler PC Sonia Stobbart for the past two weeks, was named after the police dog who died during the recent terror attacks in Paris.
"Having another puppy is hard work, there's plenty of sleepless nights and hours of your own time spent with them," she added.
"It's all worth it to know that you have a dog that you have put everything into yourself because I will spend the next eight years working with him, he will be my protector and best friend."

At Â£233bn the new stealth fighter is the most expensive weapons order ever placed, due to fly with the RAF and on the new British Navy carriers.
British test pilots have already been putting it through its paces, but the hotly awaited first appearance in UK skies was cancelled after an engine fire grounded the whole fleet.
A life size model is on display - and people queued around the block to gaze into the cockpit.
So imagine my amazement when at the back of one of the trade halls I found the stand of the Martin-Baker company - and the seat of an F35, which they were happy for visitors to try out for size.
This is the real thing, not a mock-up, as fitted to the amazing planes now flying in the States, if not here.
Martin-Baker are a company based in Buckinghamshire. They are famous around the world, but still family owned, and test their products in nearby Oxfordshire.
They started making aeroplanes after the war until Captain Valentine Baker was killed in a flying accident. His partner Sir James Martin dedicated the company to finding a way of enabling pilots to escape an aircraft safely in an emergency.
Since then they've saved an amazing 7,500 lives as airmen were able to escape their doomed planes.
It's always a last resort - but one-in-10 of the seats they have made has been used for a successful ejection.
The latest design is fully microprocessor controlled.  It is not straightforward firing out of a jet which could be flying at high speed thousands of feet up or stationary on the ground when disaster strikes. The new seat incorporates side air bags to protect the pilot's head as he flies through the canopy.
Martin-Baker seats are acknowledged throughout aviation as top quality, at the cutting edge, yet the company is still run by the two engineer sons of the founder.
It's an example of all the small firms in the UK working on what amounts to nearly 15% of the total value of the F35.
In fact everything the pilot touches is Made in Britain.
And it's a win for the Chancellor - with as much money projected to come back in tax revenue from manufacturers as the MoD has spent on its share of the order.
So I can say I've sat in the seat of a world beater. OK the rest of the plane was missing, but knowing the history of that seat made it a huge thrill nonetheless.
North Wales Police has recruited two keen and eager new members - crime-fighting puppies Luna and Diesel.
[NEXT_CONCEPT]
A major disappointment for the crowds flocking to Hampshire for the Farnborough Airshow was the non-appearance of the new American-built F35 fighter plane.