With a big dog like Loke, I don't think you'd want to put him in a trailer and tow him yourself, it would be a lot of work!
On the other hand, with Django it's not so bad, since there's two adults to pull him along, and he obviously runs alongside the trike some of the time.
As I recall, most dogs are not going to be able to run with a bike or trike for much more than 20 miles, and probably only at a moderate pace then, so if you're going to tour with a dog, you've really got to have some way to carry him or her, unless you're planning on a really low mileage tour.
I'm a little suspicious of hitches to attach dogs to bikes. Generally I think the argument against having a dog on a normal lead when cycling is that you are not in full control of your vehicle, should the dog decided to go a different route to you. Arguably your dog is trained well enough not to do this, in which case it doesn't actually need to be a on a lead.
I know that some of these hitches claim that they are designed to allow the dog to go a bit off course, but I've still got my doubts that if a big dog, like either of those two, wants to go a different direction to you, that any give in the hitch is going to be sufficient to stop things go very askew very fast.
With dogs like Loke and Django attached to recumbent trikes, I think that argument is not such an issue, since it would take a lot of effort for them to pull the trike across, and probably impossible to tip it over, unlike a traditional upright safety bicycle.
So, you need a trailer and a recumbent to pull it.
I loved this shot of Loke:
(If that doesn't appear it should be visible here).